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THE 


SENSATIONS    OF    TONE 


Bibliographical  Note. 

First  English  Edition,  June,  1875  ; 

Second  Edition,  revised  and  Aj^pendix  added,  August,  1885; 

Third  Edition,  reprinted  from  Second  Edition,  June,  1895. 


ON     THE 


SENSATIONS    OF    TONE 


AS    A    PHYSIOLOGICAL    BASIS    FOR    THE 


THEOEY      OF      MUSIC 


BY 

HERMANN  L.  F.  HELMHOLTZ,  M.D. 

LATE    FOREIGN    MEMBEE    OF    THE    ROYAL     SOCIETIES    OF    LONDON    AND    EDINBURGH, 

PROFESSOR    OF    PHYSIOLOGY    IN    THE    UNIVERSITY    OF    HEIDELBERG,    AND 

PROFESSOR    OF    PHYSICS    IN    THE    UNIVERSITY    OF    BERLIN 


Trmisluted,  thoroughly  Revised  ami   Corrected,  rendered   conformable  to   the  Fourth 

(a7id  last)  German  Edition  of  1877,  with  numerous  additional  Notes  and  a 

New  additional  Appendix  bringing  doicn  information  to  1885 

and  especially  adapted  to  the  use  of  3Iusical  Students 

BY 

ALEXANDER    J.    ELLIS 

B.A.,   F.R.S.,   F.S.A.,   F.C.P.S.,   F.C.P. 

T^\^CE    PRESIDENT    OF    THE    PHILOLOGICAL    SOCIETY,   MEMBER    OF    THE    MATHEMATICAL    SOCIETY, 

FORMERLY    SCHOLAR   OF   TRINITY   COLLEGE,    CAMBRIDGE, 
AUTHOR    OF    '  EARLY  ENGLISH  PRONUNCIATION  '    AND    '  ALGEBRA  IDENTIFIED  WITH   GEOMETRY  ' 


THIRD   £DITPON 


LONDON 
LONGMANS,     GREEN,     AND     CO. 

AND   NEW   YORK 
1895 

All    rights    reserved 


PHYSICS  DEPT. 


(JIL.^^Ua^<$>^  ' 


ABEEDEEN    UNIVERSITY    PRESS. 


TRANSLATOR'S    NOTICE 

TO    THE 

SECOND  ENGLISH  EDITION. 

Ix  preparing  a  new  edition  of  this  translation  of  Professor  Helmlioltz's  great  work  on 
the  Sensations  of  Tone,  which  was  originally  made  from  the  third  German  edition 
of  1870,  and  was  finished  in  June  1875,  my  first  care  was  to  make  it  exactly 
conform  to  i\\e  fon,rth  German  edition  of  1877  (the  last  which  has  appeared).  The 
numerous  alterations  made  in  the  fourth  edition  are  specified  in  the  Author's  pre- 
face. In  order  that  no  merely  verbal  changes  might  escape  me,  every  sentence 
of  my  translation  was  carefully  re-read  with  the  German.  This  has  enabled  me 
to  correct  several  misprints  and  mistranslations  which  had  escaped  my  previous 
very  careful  revision,  and  I  have  taken  the  opportunity  of  improving  the  language 
in  many  places.     Scarcely  a  page  has  escaped  such  changes. 

Professor  Helmholtz's  book  having  taken  its  place  as  a  work  which  all  candidates 
for  musical  degrees  are  expected  to  study,  my  next  care  was  by  supplementary 
notes  or  brief  insertions,  always  carefully  distinguished  from  the  Author's  by  being 
inclosed  in  [  ],  to  explain  any  difficulties  which  the  student  might  feel,  and  to  shew 
him  how  to  acquire  an  insight  into  the  Author's  theories,  which  were  quite  strange 
to  musicians  when  they  appeared  in  the  first  German  edition  of  1863,  but  in  the 
twenty-two  years  which  have  since  elapsed  have  been  received  as  essentially  valid 
by  those  competent  to  pass  judgment. 

For  this  purpose  I  have  contrived  the  Harmonical,  explained  on  pp.  466-469, 
by  winch,  as  shewn  in  numerous  footnotes,  almost  every  point  of  theory  can  be 
illustrated  ;  and  I  have  arranged  for  its  being  readily  procurable  at  a  moderate 
charge.  It  need  scarcely  be  said  that  my  interest  in  this  instrument  is  purely 
scientific. 

My  own  Appendix  has  been  entirely  re-written,  much  has  been  rejected  and  the 
rest  condensed,  but,  as  may  be  seen  in  the  Contents,  I  have  added  a  considerable 
amount  of  information  about  points  hitherto  little  known,  such  as  the  Determi- 
nation and  History  of  Musical  Pitch,  Non-Harmonic  scales,  Tuning,  &c.,  and  in 
especial  I  have  given  an  account  of  the  work  recently  done  on  Beats  and  Com- 
binational Tones,  and  on  Vowel  Analysis  and  Synthesis,  mostly  since  the  fourth 
German  edition  appeared. 

Finally,  I  wish  gratefully  to  acknowledge  the  assistance,  sometimes  very  great, 
which  I  have  received  from  Messrs.  D.  J.  Blaikley,"R.  H.  M.  Bosanquet,  Colin 
Brown,  A.  Cavaille-Coll,  A.  J.  Hipkins,  W.  Huggins,  F.R.S.,  Shuji  Isawa,  H. 
Ward  Poole,  R.  S.  Rockstro,  Hermann  Smith,  Steinway,  Augustus  Stroh,  and 
James  Paid  White,  as  will  be  seen  by  referring  to  their  names  in  the  Index. 


ALEXANDER  J.  ELLIS. 


25  Argyll  Road,  Kensington 
July,  1885. 


238213 


AUTHOR'S    PREFACE 

TO    THE 

FIRST  GERMAN  EDITION. 


In  laying  before  the  Public  the  result  of  eight  years'  labour,  I  must  first  pay  a 
debt  of  gratitude.  The  following  investigations  could  not  have  been  accomplished 
without  the  construction  of  new  instruments,  which  did  not  enter  into  the  inventory 
of  a  Physiological  Institute,  and  which  far  exceeded  in  cost  the  usual  resources  of 
a  German  philosopher.  The  means  for  obtaining  them  have  come  to  me  from 
unusual  sources.  The  apparatus  for  the  artificial  construction  of  vowels,  described 
on  pp.  121  to  126,  I  owe  to  the  munificence  of  his  Majesty  King  Maximilian  of 
Bavaria,  to  Avhom  German  science  is  indebted,  on  so  many  of  its  fields,  for  ever- 
ready  sympathy  and  assistance.  For  the  construction  of  my  Harmonium  in 
perfectly  natural  intonation,  descriljcd  on  p.  316,  I  was  able  to  use  the  Soemmering 
prize  which  had  been  awarded  me  by  the  Senckenberg  Physical  Society  {die 
Senckenbergische  nahirforscheiide  Gesellschaft)  at  Frankfurt-on-the-Main.  While 
publicly  repeating  the  expression  of  my  gratitude  for  this  assistance  in  my  investi- 
gations, I  hope  that  the  investigations  themselves  as  set  forth  in  this  book  will 
prove  far  better  than  mere  words  how  earnestly  I  have  endeavoured  to  make  a 
worthy  use  of  the  means  thus  placed  at  my  conmiand. 

H.   HELMHOLTZ. 
Heidelberg  :  October,  1862. 


AUTHOR'S    PREFACE 

TO   THE 

THIRD  GERMAN  EDITION. 

The  present  Third  Edition  has  been  much  more  altered  in  some  parts  than  the 
second.  Thus  in  the  sixth  chapter  I  have  been  able  to  make  use  of  the  new 
physiological  and  anatomical  researches  on  the  ear.  This  has  led  to  a  modification 
of  my  view  of  the  action  of  Corti's  ai'ches.  Again,  it  appears  that  the  pecviliar 
articulation  between  the  auditory  ossicles  called  'hammer'  and  'anvil'  might  easily 
cause  within  the  ear  itself  the  formation  of  harmonic  upper  partial  tones  for  simple 
tones  which  are  sounded  loudly.  By  this  means  that  pecidiar  series  of  upper  partial 
tones,  on  the  existence  of  which  the  present  theory  of  music  is  essentially  founded, 
receives  a  new  subjective  value,  entirely  independent  of  external  alterations  in 
the  quality  of  tone.  To  illustrate  the  anatomical  descriptions,  I  have  been  able 
to  add  a  series  of  new  woodcuts,  principally  from  Henle's  Manual  of  Anatomy, 
with  the  author's  permission,  for  which  I  here  take  the  opportunity  of  publicly 
thankine:  him. 


PREFACE.  vii 

1  have  made  many  changes  iu  re-editing  the  section  on  the  History  of  Music, 
and  hope  that  I  have  improved  its  connection.  I  must,  however,  request  the 
reader  to  regard  this  section  as  a  mere  compilation  from  secondaiy  sources  ;  I 
have  neither  time  nor  preliminary  knowledge  sufficient  for  original  studies  in  this 
extremely  difficult  field.  The  older  history  of  music  to  the  commencement  of 
Discant,  is  scarcely  more  than  a  confused  heap  of  [secondary  subjects,  while  we 
can  only  make  hypotheses  concerning  the  principal  matters  in  question.  Of 
course,  however,  every  theoi-y  of  music  must  endeavour  to  bi-ing  some  order  into 
this  chaos,  and  it  cannot  be  denied  that  it  contains  many  important  facts. 

For  the  representation  of  pitch  in  just  or  natural  intonation,  I  have  abandoned 
the  method  originally  proposed  by  Hauptmann,  which  was  not  sufficiently  clear  in 
involved  cases,  and  have  adopted  the  system  of  Herr  A.  von  Oettingen  [p.  276] , 
as  had  already  been  done  in  M.  G.  Gueroult's  French  translation  of  this  book. 

[A  comparison  of  the  Third  with  the  Second  editions,  shewing  the  changes  and  additions 
individually,  is  here  omitted.] 

If  1  may  be  allowed  in  conclusion  to  add  a  few  words  on  the  reception  expe- 
rienced by  the  Theory  of  Music  here  propounded,  I  should  say  that  published 
objections  almost  exclusively  relate  to  my  Theory  of  Consonance,  as  if  this  were 
the  pith  of  the  matter.  Those  who  prefer  mechanical  explanations  express  their 
regret  at  my  having  left  any  room  in  this  field  for  the  action  of  artistic  invention 
and  esthetic  inclination,  and  they  have  endeavoured  to  complete  my  system  by 
new  numerical  speculations.  jT)*'^®^'  critics  with  more  metaphysical  proclivities 
have  rejected  my  Theory  of  Consonance,  and  with  it,  as  they  imagine,  my  whole 
Theory  of  Music,  as  too  coarsely  mechanical.  ; 

T  hope  my  critics  will  excuse  me  if  I  conclude  from  the  opposite  nature  of 
their  objections,  that  I  have  struck  out  nearly  the  right  path.  As  to  my  Theory 
of  Consonance,  I  must  claim  it  to  be  a  mere  systematisation  of  observed  facts 
(with  the  exception  of  the  functions  of  the  cochlea  of  the  ear,  which  is  moreover 
an  hypothesis  that  may  be  entirely  dispensed  with).  But  I  consider  it  a  mistake 
to  make  the  Theory  of  Consonance  the  essential  foundation  of  the  Theory  of 
Music,  and  I  had  thought  that  this  opinion  was  clearly  enough  expressed  in  my  book. 
The  essential  basis  of  Music  is  Melody.  Harmony  has  become  to  \Vestem  Euro- 
peans during  the  last  three  centuries  an  essential,  and,  to  our  present  taste, 
indispensable  means  of  strengthening  melodic  relations,  but  finely  developed 
music  existed  for  thousands  of  years  and  still  exists  in  ultra-European  nations, 
without  any  hannony  at  all.  And  to  my  metaphysico-esthetical  opponents  I  must 
reply,  that  I  cannot  think  I  have  undervalued  the  artistic  emotions  of  the  human 
mind  in  the  Theory  of  Melodic  Constmction,  by  endeavouring  to  establish  the 
physiological  facts  on  which  esthetic  feeling  is  based.  But  to  those  who  think  I 
have  not  gone  far  enough  in  my  physical  explanations,  1  answer,  that  in  the  first 
place  a  natural  philosopher  is  never  bound  to  construct  systems  about  everything  he 
knows  and  does  not  know  ;  and  secondly,  that  I  should  consider  a  theory  which 
claimed  to  have  shewn  that  all  the  laws  of  modern  Thorough  Bass  were  natural 
necessities,  to  stand  condemned  as  having  proved  too  much. 

Musicians  have  found  most  fault  with  the  manner  in  which  I  have  characterised 
the  Minor  Mode.  I  must  refer  in  reply  to  those  very  accessible  documents,  the 
musical  compositions  of  a.d.  1500  to  a.d.  1750,  dm-ing  v/hich  the  modern  Minor 
was  developed.  These  will  shew  how  slow  and  fluctuating  was  its  development, 
and  that  the  last  traces  of  its  incomplete  state  are  still  visible  in  the  works  of 
Sebastian  Bach  and  Handel. 


Heidelberg  :  May,  1870. 


AUTHOR'S    PREFACE 


FOURTH   GERMAN    EDITION. 


In  the  essential  conceptions  of  musical  relations  I  have  found  nothing  to  alter  in 
this  new  edition.  In  this  respect  I  can  but  maintain  what  I  have  stated  in  the 
chapters  containing  them  and  in  my  preface  to  the  third  [German]  edition.  In 
details,  however,  much  has  been  remodelled,  and  in  some  parts  enlarged.  As  a 
guide  for  readers  of  former  editions,  I  take  the  liberty  to  enumerate  the  following 
places  containing  additions  and  alterations.* 

P.  16d,  note*. — On  the  French  system  of  counting  vibrations. 

P.  18«. — Appunn  and  Preyer,  limits  of  the  highest  audible  tones. 

Pp.  596  to  65b. — On  the  circumstances  under  which  we  distinguish  compound  sensations. 

P.  16a,  b,  c. — Comparison  of  the  upper  partial  tones  of  the  strings  on  a  new  and  an  old 
grand  pianoforte. 

P.  83,  note  f. — Herr  Clement  Neumann's  observations  ou  the  vibrational  form  of  nolin 
strings. 

Pp.  89ft  to  93&.— The  action  of  blowing  organ-pipes. 

P.  1106.— Distinction  of  Ou  from  U. 

Pp.  1116  to  116a. — The  various  modifications  in  the  sounds  of  vowels. 

P.  145a. — The  ampulla?  and  semicircular  canals  no  longer  considered  as  parts  of  the  organ 
of  hearing. 

P.  1476. — Waldeyer's  and  Preyer's  measurements  adopted. 

Pp.  1506  to  151d. — On  the  parts  of  the  ear  which  perceive  noise. 

P.  1596. — Koenig's  observations  on  combinational  tones  with  tuning-forks. 

P.  176d,  note. — Preyer's  observations  on  deepest  tones. 

P.  179c.— Preyer's  observation  on  the  sameness  of  the  quality  of  tones  at  the  highest  pitches. 

Pp.  203t;  to  204«. — Beats  between  upper  partials  of  the  same  compound  tone  condition  the 
preference  of  musical  tones  with  hannonic  upper  partials. 

Pp.  328c  to  3296.  — Division  of  the  Octave  into  53  degi-ees.     Bosanquet's  harmonivmi. 

Pp.  338c  to  3396. — j\Iodulations  tluough  chords  composed  of  two  major  Thirds. 

P.  365,  note  t. — Oettingen  and  Riemann's  theory  of  the  minor  mode. 

P.  372. — Improved  electro-magnetic  driver  of  the  siren. 

P.  373ft. — Theoretical  formulte  for  the  pitch  of  resonators. 

P.  374c. — Use  of  a  soap-bubble  for  seeing  vibrations. 

Pp.  389*:^  to  3966. — Later  use  of  striking  reeds.     Theory  of  the  blowing  of  pipes. 

Pp.  403c  to  4056. — Theoretical  treatment  of  svmpathetic  resonance  for  noises. 

P.  417f^. — A.  Mayer's  experiments  on  the  audibility  of  vibrations. 

P.  428c.  d. — Against  the  defenders  of  tempered  intonation. 

P.  429. — Plan  of  Bosanquet's  Harmonium. 

H.   HELMHOLTZ. 
Berlin  :  A2Jril,  1877. 

*  [The  pages  of  this  edition  are  substituted  first  edition  of  this  translation  are  mostly 
for  the  German  throughout  these  prefaces,  pointed  out  in  footnotes  as  they  arise. — Trans- 
and  omissions   or  alterations  as  respects  the       lator.] 


CONTENTS. 


and  notes  in  [  ]  are  due  to  the  Translator,  and  the  Author  is  in  no 
way  responsible  for  their  contents. 


Translatoe's  Notice  to  the  Second  English  Edition,  p.  v. 

Author's  Preface  to  the  First  German  Edition,  p.  vi. 

Author's  Preface  to  the  Third  German  Edition,  pp.  vi-vii. 

Author's  Preface  to  the  Fourth  German  Edition,  p.  viii. 

Contents,  p.  ix. 

List  of  Figures,  p.  xv. 

List  of  Passages  in  Musical  Notes,  p.  xvi. 

List  of  Tables,  p.  xvii. 

INTRODUCTION,  pp.  1-6. 

Relation  of  Musical  Science  to  Acoustics,  1 

Distinction  between  Physical  and  Physiological  Acoustics,  3 

Plan  of  the  Investigation,  4 


PAET  I.   (pp.    7-151.) 

ON    THE    COMPOSITION    OF    VIBRATIONS. 

Upl^er  Partial  Tones,  and  Qualities  of  Tone. 

CHAPTEK  I.     On  the  Sensation  of  Sound  in  General,  pp.  8-25. 

Distinction  between  Noise  and  Musical  Tone,  8 

Musical  Tone  due  to  Periodic,  Noise  to  non-Periodic  Llotions  in  the  air,  8 

General  Property  of  Undulatory  Motion :  while  Waves  continually  advance,  the  Particles 

of  the  Medium  through  which  they  pass  execute  Periodic  ]\Iotions,  9 
Differences  in  Musical  Tones  due  to  Force,  Pitch,  and  Quality,  10 
Force  of  Tone  depends  on  Amplitude  of  Oscillation,  Pitch  on  the  length  of  the  Period  of 

Oscillation,  10-14 
Simple  relations  of  Vibrational  Numbers  for  the  Consonant  Intervals,  14 
Vibrational  Numbers  of  Consonant  Intervals  calculated  for  the  whole  Scale,  17 
Quality  of  Tone  must  depend  on  Vibrational  Form,  19 
Conception  of  and  Graphical  Representation  of  Vibrational  Form,  20 
Harmonic  Upper  Partial  Tones,  22 
Terms  explained :  Tone,  Musical  Tone,  Simple  Tone,  Partial  Tone,  Compound  Tone,  Pitch 

of  Compound  Tone,  23 

CHAPTEE  11.     On  the  Composition  of  Vibrations,  pp.  25-36. 

Composition  of  Vv'aves  illustrated  by  waves  of  water,  25 

The  Heights  of  superimposed  Waves  of  Water  are  to  be  added  algebraically,  27 

Corresponding  Superimposition  of  Waves  of  Sound  in  the  air,  28 


CONTENTS. 

A  Composite  Mass  of  Musical  Tones  will  give  rise  to  a  Periodic  Vibration  wlien  their  Pitch 

Numbers  are  Multiples  of  the  same  Number,  30 
Every  such   Composite   Mass   of   Tones  may  be   considered  to  be  composed  of    Simple 

Tones,  33 
This  Composition  corresponds,  according  to  G.  S.  Ohm,  to  the  Composition  of  a  Musical 

Tone  from  Simple  Partial  Tones,  33 


CHAPTER  III.      Analysis  of  Musical  Tones  by  Sympathetic  Ee- 
SONANCE,  pp.  36-49. 

Explanations  of  the  Mechanics  of  Sympathetic  Vibration,  3G 

Sympathetic  Resonance  occurs  when  the  exciting  vibrations  contain  a  Simple  Vibration 

corresponding  to  one  of  the  Proper  Vibrations  of  the  Sympathising  Body,  33 
Difference  in  the  Sympathetic  Resonance  of  Tuning-forks  and  Membranes,  40 
Description  of  Resonators  for  the  more  accurate  Analysis  of  Musical  Tones,  43 
Sympathetic  Vibration  of  Strings,  45 
Objective  Existence  of  Partial  Tones,  48 


CHAPTEE  IV.     On  the  Analysis  of  Musical  Tones  by  the  Ear, 
pp.  49-65. 

Slethods  for  observing  Upper  Partial  Tones,  49 

Proof  of  G.  S.  Ohm's  Law  by  means  of  the  tones  of  Plucked  Strings,  of  the  Simple  Tones 
of  Tuning-forks,  and  of  Resonators,  51 

Difference  between  Compound  and  Simple  Tones,  56 

Seebeck's  Objections  against  Ohm's  Law,  58 

The  Difficulties  experienced  in  perceiving  Upper  Partial  Tones  analytically  depend  upon  a 
peculiarity  common  to  all  human  sensations,  59 

We  practise  observation  on  sensation  only  to  the  extent  necessary  for  clearly  apprehend- 
ing the  external  world,  62 

Analysis  of  Compound  Sensations,  63 


CHAPTEE   Y.      On  the  Differences  in  the  Quality  of  Musical 
Tones,  pp.  65-119. 

Noises  heard  at  the  beginning  or  end  of  Tones,  such  as  Consonants  in  Speech,  or  during 
Tones,  such  as  Wind-rushes  on  Pipes,  not  included  in  the  Musical  Quality  of  Tone, 
which  refers  to  the  uniformly  continuous  musical  sound,  65 

Limitation  of  the  conception  of  Musical  Quality  of  Tone,  68 

Investigation  of  the  Upper  Partial  Tones  which  are  present  in  different  Musical  Qualities 
of  Tone,  69 

1.  ]\iusical  Tones  without  Upper  Partials,  69 

2.  Musical  Tones  with  Inharmonic  Upper  Partials,  70 

3.  Musical  Tones  of  Strings,  74 

Strings  excited  by  Striking,  74 

Theoretical  Intensity  of  the  Partial  Tones  of  Strings,  79 

4.  ]\Iusical  Tones  of  Bowed  Instruments,  80 

5.  :Musical  Tones  of  Flute  or  Flue  Pipes,  88 

6.  ilusical  Tones  of  Reed  Pipes,  95 

7.  Vowel  Qualities  of  Tone,  103 

Results  for  the  Character  of  Musical  Tones  in  general,  118 


CHAPTEE  YI.      On    the    Apprehension   of    Qualities    of    Tone, 
pp.  119-151. 

Does  Quality  of  Tone  depend  on  Difference  of  Phase  ?  119 

Electro-magnetic  Apparatus  for  answering  this  question,  121 

Artificial  Vowels  produced  by  Tuning-forks,  123 

How  to  produce  Difference  of  Phase,  125 

Musical  Quality  of  Tone  independent  of  Difference  of  Phase,  126 

Artificial  Vowels  produced  by  Organ  Pipes,  128 

The  Hypothesis  that  a  Series  of  S}-mpathetical  Vibrators  exist  in  the  ear,  explains  its 
peculiar  apprehension  of  Qualities  of  Tone,  129 

Description  of  the  parts  of  the  internal  ear  which  are  capable  of  vibrating  sympa- 
thetically, 129 

Damping  of  Vibrations  in  the  Ear,  142 

Supposed  Function  of  the  Cochlea,  145 


CONTENTS. 


PAKT    11.    (pp.    152-283.) 

ON    THE    INTERRUPTIONS    OF     HARMONY. 

Gonihinatiomil   Tones  and  Beats,   Consonance  and  Dissonance. 

CHAPTEE  VII.     Combinational  Tones,  pp.  152-159. 

Combinational  Tones  arise  when  Vibrations  which  are  not  of   infinitesimal  magnitude  are 

combined,  152 
Description  of  Combinational  Tones,  153 
Law  determining  their  Pitch  Numbers,  254 
Combinational  Tones  of  different  orders,  155 

Difference  of  the  strength  of  Combinational  Tones  on  different  instruments,  157 
Occasional  Generation  of  Combinational  Tones  in  the  ear  itself,  158 

CHAPTEE  VIII.     On  the  Beats  of  Simple  Tones,  pp.  159-173. 

Interference  of  Two  Simple  Tones  of  the  same  pitch,  160 

Description  of  the  Polyphonic  Siren,  for  experiments  on  Interference,  161 

Eeinforcement  or  Enfeeblement  of  Sound,  due  to  difference  of  Phase,  163 

Interference  gives  rise  to  Beats  when  the  Pitch  of  the  two  Tones  is  slightly  different,  164 

Law  for  the  Number  of  Beats,  165 

Visible  Beats  on  Bodies  vibrating  sympathetically,  166 

Limits  of  Kapidity  of  Audible  Beats,  1G7 

CHAPTEE  IX.     Deep  and  Deepest  Tones,  pp.  174-179. 

Former  Investigations  were  insufficient,  because  there  was  a  possibility  of  the  ear  being 
deceived  by  Upper  Partial  Tones,  as  is  shewn  by  the  number  of  Beats  on  the  Siren,  174 

Tones  of  less  than  thirty  Vibrations  in  a  second  fall  into  a  Drone,  of  which  it  is  nearly 
or  quite  impossible  to  determine  the  Pitch,  175 

Beats  of  the  Higher  Upper  Partials  of  one  and  the  same  Deep  Compound  Tone,  178 

CHAPTEE  X.     Beats  of  the  Uppee  Partial  Tones,  pp.  179-197. 

Any  two  Partial  Tones  of  any  two  Compound  Tones  may  beat  if  they  are  sufficiently 
near  in  pitch,  but  if  they  are  of  the  same  pitch  there  will  be  consonance,  179 

Series  of  the  different  Consonances,  in  order  of  'the  Distinctness  of  their  Delimitation,  183 

Number  of  Beats  which  arise  from  Mistuning  Consonances,  and  their  effect  in  producing 
Roughness,  184 

Disturbance  of  any  Consonance  by  the  adjacent  Consonances,  186 

Order  of  Consonances  in  respect  to  Harmoniousness,  188 

CHAPTEE  XL     Beats  due  to  Combinational  Tones,  pp.  197-211. 

The  Differential  Tones  of  the  first  order  generated  by  two  Partial  Tones  are  capable  of 
producing  very  distinct  beats,  197 

Differential  Tones  of  higher  orders  produce  weaker  beats,  even  in  the  case  of  simple  gene- 
rating tones,  199 

Influence  of  Quality  of  Tone  on  the  Harshness  of  Dissonances  and  the  Harmoniousness 
of  Consonances,  205 

CHAPTEE  XII.     Chords,  pp.  211-233. 

Consonant  Triads,  211 

Major  and  Minor  Triads  distinguished  by  their  Combinational  Tones,  214 
Relative  Harmoniousness  of  Chords  in  different  Inversions  and  Positions,  218 
Retrospect  on  Preceding  Investigations,  226 


CONTENTS. 


PAKT   III.     (pp.   234-371.) 

THE    RELATIONSHIP    OF    MUSICAL    TONES. 

Scales  and   Tonality. 

CHAPTEE  XIII.  General  View  of  the  Different  Principles 
OF  Musical  Style  in  the  Development  of 
Music,  pp.  234-249. 

Difference  between  the  Physical  and  the  Esthetical  Method,  234 

Scales,  Keys,  and  Harmonic  Tissues  depend  upon  esthetic  Principles  of  Style  as  well  as 

Physical  Causes,  235 
Illustration  from  the  Styles  of  Architecture,  235 
Three  periods  of  Music  have  to  be  distinguished,  236 

1.  Homophonic  Music,  237 

2.  Polyphonic  ]\Iusic,  244 

3.  Harmonic  Music,  246 

CHAPTEE  XIV.     The  Tonality  of  Homophonic  Music,  pp.  250-290. 

Esthetical  Reason  for  Progression  by  Intervals,  250 

Tonal  Relationship  in  IMelody  depends  on  the  identity  of  two  partial  tones,  253 

The  Octave.  Fifth,  and  Fourth  were  thus  first  discovered,  253 

Variations  in  Thirds  and  Sixths,  255 

Scales  of  Five  Tones,  used  by  Chinese  and  Gaels,  258 

The  Chromatic  and  Enharmonic  Scales  of  the  Greeks,  262 

The  Pythagorean  Scales  of  Seven  tones,  266 

The  Greek  and  Ecclesiastical  Tonal  ]\Iodes,  267 

Early  Ecclesiastical  Modes,  272 

The  Rational  Construction  of  the  Diatonic  Scales  by  the  principle  of  Tonal  Relationship  in 

the  first  and  second  degrees  gives  the  five  Ancient  ^Melodic  Scales,  272 
Introduction  of  a  more  Accurate  Notation  for  Pitch,  276 

Peculiar  discovery  of  natural  Thirds  in  the  Arabic  and  Persian  Tonal  Systems,  280 
The  meaning  of  the  Leading  Note  and  consequent  alterations  in  the  Modern  Scales,  285 

CHAPTEE  XV.     The  Consonant  Chords  of  the  Tonal  Modes,  pp. 
290-309. 

Chords  as  the  Representatives  of  compound  Musical  Tones  with  peculiar  qualities,  290 
Reduction  of  aU  Tones  to  the  closest  relationship  in  the  popular  harmonies  of  the  Manor 
Mode,  292  ^  i-  i-  .  j 

Ambiguity  of  Iifinor  Chords,  294 

The  Tonic  Chord  as  the  centre  of  the  Sequence  of  Chords,  296 

Relationship  of  Chords  of  the  Scale,  297 

The  ]\Iajor  and  INIinor  IModes  are  best  suited  for  Harmonisation  of  all  the  Ancient  Modes, 

298 
Modern  Remnants  of  the  old  Tonal  IModes,  306 

CHAPTEE  XVI.     The  System  of  Keys,  pp.  310-330. 

Relative  and  Absolute  Character  of  the  different  Keys,  310 
Modulation  leads  to  Tempering  the  Intonation  of  the  Intervals,  312 

Hauptmann's  System  admits  of  a  Simplification  vfhich  makes  its  Realisation  more  Practi- 
cable, 315 
Description  of  an  Harmonium  with  Just  Intonation,  316 
Disadvantages  of  Tempered  Intonation,  322 
Modulation  for  Just  Intonation,  327 


CONTENTS. 
CHAPTEE  XVII.     Of  Discords,  pp.  330-350. 

Envuneration  of  the  Dissonant  Intervals  in  the  Scale,  331 

Dissonant  Triads,  338 

Chords  of  the  Seventh,  341 

Conception  of  the  Dissonant  Note  in  a  Discord,  346 

Discords  as  representatives  of  compound  tones,  347 


CHAPTEE  XVIII.     Laws  of  Progression  of  Parts,  pp.  350-362. 

Tlie  IMusical  Connection  of  the  Notes  in  a  INIelody,  350 

Consequent  Rules  for  the  Progression  of  Dissonant  Notes,  353 

Resolution  of  Discords,  354 

Choral  Sequences  and  Resolution  of  Chords  of  the  Seventh,  355 

Prohibition  of  Consecutive  Fifths  and  Octaves,  369 

Hidden  Fifths  and  Octaves,  3G1 

False  Relations,  361 


CHAPTEE  XIX.     EsTHETicAL  Eelations,  pp.  362-371. 

Review  of  Results  obtained,  362 

Law  of  Unconscious  Order  in  Works  of  Art,  366 

The  Law  of  IMelodic  Succession  depends  on  Sensation,  not  on  Consciousness,  368 

And  similarly  for  Consonance  and  Dissonance,  869 

Conclusion,  371 


APPENDICES,  pp.  327-556. 

I.  On  an  Electro-Magnetic  Driving  IMachine  for  the  Siren,  372 

II.  On  the  Size  and  Construction  of  Resonators,  372 

III.  On  the  Motion  of  Plucked  Strings,  374 

IV.  On  the  Production  of  Simple  Tones  by  Resonance,  377 
V.  On  tlie  Vibrational  Forms  of  Pianoforte  Strings,  380 

VI.     Analysis  of  the  ]\Iotion  of  Violin  Strings,  384 
VII.     On  the  Theory  of  Pipes,  388 

A.  Influence  of  Resonance  on  Reed  Pipes,  388 

B.  Theory  of  the  Blowing  of  Pipes,  390 

I.  The  Blowing  of  Reed  Pipes,  390 

II.  The  Blowing  of  Flue  Pipes,  394 
[Additions  by  Translator,  396] 

VIII.     Practical  Directions  for  Performing  the  Experiments  on  the  Composition  of  Vowels, 
398 
IX.     On  the  Phases  of  Waves  caused  by  Resonance,  400 
X.     Relation  between  the  Strength  of  Sympathetic  Resonance  and  the  Length  of  Time 

required  for  the  Tone  to  die  away,  405 
XL     Vibrations  of  the  Wembrana  Basilaris  in  the  Cochlea,  406 
XII.     Theory  of  Combinational  Tones,  411 

XIII.  Description  of  the  Mechanism  employed  for  opening  the  several  Series  of  Holes  in 

the  Polyphonic  Siren,  413 

XIV.  Variation  in  the  Pitch  of  Simple  Tones  that  Beat,  414 

XV.     Calculation  of  the  Intensity  of  the  Beats  of  Different  Intervals,  415 
XVI.     On  Beats  of  Combinational  Tones,  and  on  Combinational  Tones  in  the  Siren  and 

Harmonium,  418 
XVII.     Plan  for  Justly-Toned  Instruments  with  a  Single  Manual,  421 
XVIII.     Just  Intonation  in  Singing,  422 
XIX.     Plan  of  Mr.  Bosanquet's  Manual,  429 
[XX.     Additions  by  the  Translator,  430-556 

*»*  See  separate  Tables  of  Contents  prefixed  to  each  Section. 
[Sect.  A.  On  Temperament,  430 
[Sect.  B.  On  the  Determination  of  Pitch  Numbers,  441 


CONTENTS. 

[App.  XX.  Additions  by  the  Tra.nsla,toi-—coniinued. 

*^*  See  separate  Tables  of  Contents  prefixed  to  each  Section. 
[Sect.  C.  On  the  Calculation  of  Cents  from  Interval  Ratios,  446 
[Sect.  D.  Musical  Intervals,  not  exceeding  an  Octave,  arranged  in  order  of  Width 
451  ' 

[Sect.  E.  On  Musical  Duodenes,  or  the  Development  of  Just  Intonation   for 

Harmony,  457 
[Sect.  P.   Experimental  Instruments  for  exhibiting  the  effects  of  Just  Intonation 
466  ' 

[Sect.  G.  On  Tuning  and  Intonation,  483 
[Sect.  H.  The  History  of  Musical  Pitch  in  Europe,  493 
[Sect.  K.  Non-Harmonic  Scales,  514 

[Sect.  L.  Recent  Work  on  Beats  and  Combinational  Tones,  527 
[Sect.  M.  Analysis  and  Synthesis  of  Vowel  Sounds,  538 
[Sect.  N.  Miscellaneous  Notes,  544 

[INDEX,  557-576] 


LIST   OF   FIGURES 


1.  Seebeck's  Siren,  lie 

2,  3,  4.  Cagniard  de  la  Tour's  Siren,  12b 

5.  Tuning-fork  tracing  its  Curve,  206 

6.  Curve  traced  in  Phonautograph,  20d 

7.  Curve  of  Simple  Vibration,  216 

8.  Curve  of  ]\Iotion  of  Hammer  moved  by 

Water-wheel,  2lc 

9.  Curve  of  :Motion  of  Ball  struck  up  on 

its  descent,  21c 

10.  Reproduction  of  fig.  7,  2M 

11.  Curve  shewing  the  Composition  of   a 

simple  Note  and  its  Octave  in  two 
different  phases,  306,  c 

12.  Curve  shewing  the  Composition  of  a 

simple  note  and  its  Twelfth  in  two 
different  phases,  326 

13.  Tuning-fork  on  Resonance  Box,  40a 

14.  Forms  of  Vibration  of  a  Circular  Mem- 

brane, 40f,  d 

15.  Pendulum    excited    by    a    membrane 

covering  a  bottle,  42« 

16.  a.   Spherical  Resonator,  436 
b.  Cylindi-ical  Resonator,  43f 

17.  Forms  of  Vibration  of  Strings,  46«,  6 

18.  Forms    of   Vibration  of   a  String  de- 

flected by  a  Point,  54«,  6 

19.  Action  of  such  a  String  on  a  Sounding- 

board,  54c 

20.  Bottle  and  Blow-tube  for  producing  a 

simple  Tone,  60c 

21.  Sand  figures  on  circular  elastic  plates, 

71c 

22.  The  Vibration  Microscope,  816 

23.  Vibrations   as   seen   in   the  Vibration 

Microscope,  826 
•4.  Vibrational  Forms  for  the  middle  of  a 
Viohn  String,  836 

25.  Crumples  on  the  vibrational  form  of  a 

violin  string,  846 

26.  Gradual  development  of  Octave  on  a 

violin  string  bowed  near  the  bridge, 
856 

27.  An   open   wooden   and   stopped  metal 

organ  flue-pipe,  88 

28.  Free  reed  or  Harmonium  vibrator,  956 

29.  Free  and  striking   reed   on  an  organ 

pipe  partly  in  section,  96rt,  6 

30.  IMembranous  double  reed,  97a 

31.  Reproduction  of  fig.  12,  120^,  b 

32.  Fork  with  electro-magnetic  exciter,  and 

sliding    resonance    box   with    a   lid 
(aa-tificial  vowels),  1216 


35 


33.  Fork  with  electro-magnet  to  serve  as 

interrupter  of  the  current  (artificial 
vowels),  1226 

34.  Appearance  of  figiires  seen  through  the 

vibration   microscope   by  two  forks 
when    the    phase   changes   but   the 
tuning  is  correct,  I26d 
The  same  when  the  tuning  is  slightly 
altered,  127ft 

56.  Construction  of  the  ear,  general  view, 

with  meatus   auditorius,  labyrinth, 
cochlea,  and  Eustachian  tube,  129c 

57.  The  three  auditory  ossicles,  hammer, 

anvil,  and  stirrup,  in  their  relative 
positions,  130c 

38.  Two  views  of  tlie  hammer  of  the  ear, 

1316 

39.  Left  temporal   bone  of   a  newly-born 

child  with  the  auditory  ossicles  in 
situ,  131c 

40.  Right   drumskin   with   hammer    seen 

from  the  inside,  131c 

41.  Two  views  of  the  right  anvil,  133« 

42.  Three  views  of  the  right  stirrup,  134a 

43.  A,    left   labyrinth   from   without.     B, 

right    labyrinth    from   within.      C, 
left  labyrinth  from  above,  1366,  c 

44.  Utriculus  and  membranous  semicircular 

canals  (left  side)  seen  from  without, 
137« 

45.  Bony  cochlea  (right   side)   opened   in 

front,  1.37c,  d 
Transverse   section   of    a    spire    of    a 

cochlea   which  has    been    softened 

in  hydrochloric  acid,  ISSa,  b 
Max  Schultze's  hairs  on  the  internal 

surface   of    the    epithsnum    in    the 

am^ndkc,  138c,  d 

48.  Expansion  of  the  cochlean  nerve,  139c 

49.  Corti's  membrane,  140rt,  6,  c 

50.  Corti's  rods  or  arches  separate,  140(/ 

51.  Corti's  rods  or  arches  in  situ,  1416,  c 

52.  Diagram  of  the  law  of  decrease  of  sym- 

pathetic resonance,  144c,  d 
Interference     of     similarly     disposed 

waves,  1606 
Interference   of   dissimilarly   disposed 

waves,  160c 

55.  Lines    of    silence    of     a    tuning-fork, 

161c 

56.  The  Polyphonic  Siren,  162 

57.  Diagram  of  origin  of  beats,  165ff 


46. 


47. 


53. 


54. 


XVI 


LIST  OF  PASSAGES  IN  MUSICAL  NOTES. 


58.  Phouautographic      representation     of 

beats,  166rt 

59.  Identical  with  fig.  52  but  now  taken  to 

shew  the  intensity  of  beats  excited 
by  tones  making  different  intervals, 
172c 

60.  A  and  B.     Diagram  of  the  comparative 

roughness  of  intervals  in  the  first 
and  second  octaves,  193b,  c 

61.  Diagram  of  the  roughness  of  dissonant 

intervals,  333« 

62.  Reproduction  of  fig.  24  A,  p.  385& 

63.  Diagram   of   the  motion   of    a   violin 

string,  387c 


14.  Diagram  of  the  arrangements  for  the 

experiments  on  the  composition  of 

vowels,  399b,  c 
<5.  Mechanism   for    opening    the    several 

series   of   holes   in    the    Polyphonic 

Siren,  414rt, 
16.  Section,    Elevation,  and   Plan  of  Mr. 

Bosanquet's  Manual,  429 

Th  Additions  by  Translator. 
57.  Perspective  view  of  Mr.  Colin  Brown's 

Fingerboard,  47 Id 
18.  Perspective  view,  69  plan,  70  section 

of  Mr.  H.  W.  Poole's  Keyboard,  475 


LIST  OF  PASSAGES  IN  MUSICAL  NOTES. 


The  small  octave,  15fZ 

The  once  and  twice  accented  octave,  16a,  b 

The  great  octave,  166 

The  first  16  Upper  Partials  of  C'66,  22c 

The  first  8  Upper  Partials  of  6132,  50«- 

Prof.  Helmholtz's  Vowel  Resonances,  UOb 

First  differential  tones  of  the  usual  har- 
monic interval,  1546 

Differential  tones  of  different  orders  of  the 
usual  harmonic  intervals,  1556,  c 

Summational  tones  of  the  usual  harmonic 
intervals,  156ft 

Examples  of  beating  partials,  180c 

Coincident  partials  of  the  principal  con- 
sonant intervals,  183(Z 

Coincident  converted  into  beating  partials 
by  altering  pitch  of  upper  tone,  186c 

Examples  of  intervals  in  which  a  pair  of 
partials  beat  33  times  in  a  second,  1 92« 

Major  Triads  with  their  Combinational 
Tones,  215a 

Minor  Triads  with  their  Combinational 
Tones,  2156 

Consonant  Intervals  and  their  Combina- 
tional Tones,  218c 

The  most  Perfect  Positions  of  the  Major 
Triads,  219c 

The  less  Perfect  Positions  of  the  Major 
Triads,  220c 

The  most  Perfect  Positions  of  the  Minor 
Triads,  2216 

The  less  Perfect  Positions  of  the  INIinor 
Triads,  221c 


The    most    Perfect    Positions     of     Major 

Tetrads   within   the   Compass   of   Two 

Octaves,  223c 
Best  Positions  of  Minor  Tetrads  with  their 

false  Combinational  Tones,  224« 
Ich  bin  spatziercn  gegangen,  2386 
Sic  canta  comma,  2396 
Palestrina's    Stabat    Mater,    first    4    bars, 

247c 
Chinese  air  after  Barrow,  260« 
Cockle  Shells,  older  form,  2606 
Blythe,  blythc,  and  merry  are  vx,  261ffl 
Chinese  temple  hymn  after  Bitschurin,  2616 
Braes  of  Bulqtihidder,  261c 
Five  Forms  of  Closing  Chords,  291c 
Two  complete  closes,  293c 
IMode  of  the  Fourth,  three  forms  of  com- 
plete cadence,  302(7 
Concluding  bars  of  S.  Bach's  Chorale,  Was 

viein  Gott  ivill,  das  gescheJi'  allzeit,  3046 
End  of  S.  Bach's  Hymn,   Veni  redemptor 

gentium,  305a 
Doric  cadence  from  And  with  His  stripes 

we  are  healed,  in  Handel's  Messiah,  307a 
Doric  cadence  from  Hear,  Jacob's  God,  in 

Handel's  Samson,  3076 
Examples  of  False  Minor  Triad,  340a 
Examples  of  Hidden  Fifths,  361c? 
Example  of  Duodenals,  465c 
Mr.  H.  W.  Poole's  method  of  fingering  and 

treatment  of  the  harmonic  Seventh,  477a 
Mr,  H.  W.  Poole's  Double  Diatonic  or  Di- 

chordal  Scale  in  Ci' with  accidentals,  478a 


LIST    OF    TABLES. 


Pitch  Numbers  of  Notes  in  Just  JMajor 
Scale,  17« 

[Scale  of  Haimonical,  '17c,  d] 

[Analogies  of  notes  of  the  piano  and  colours 
of  the  Spectrum,  IBd'] 

Pitch  of  the  different  forms  of  vibration 
of  a  circular  membrane,  41c 

Relative  Pitch  Numbers  of  the  prime  and 
proper  tones  of  a  red  free  at  both  ends, 
56« 

Proper  Tones  of  circular  elastic  plates,  72a 

Proper  Tones  of  Bells,  72c 

Proper  Tones  of  Stretched  Membranes,  7Sb 

Theoretical  Intensity  of  the  Partial  Tones 
of  Strings,  7i'c 

[Velocity  in  Soimd  in  tubes  of  different 
diameters — Blaikley,  9Qd] 

[Partials  of  £\)  Clarinet— Blaikley,  99c] 

[Harmonics  of  £\^  horn,  99d] 

[Compass  of  Eegisters  of  male  and  female 
voices — Behnke,  ]  01  d] 

Vowel  trigram — Du  Bois  Raymond,  senior, 
106& 

Vowel  Resonances  according  to  Helmholtz 
and  Donders,  l(i9b 

[Vowel  Resonances  according  to  (1)  Reyher, 
[i)  Hellwag,  (3)  Florcke,  (4)  Donders  after 
Helmholtz,  (5)  Dondeis  after  Merkel, 
(6)  Helmholtz,  (7)  Merkel,  (8)  Koenig.. 
(9)  Trautmann,  109rf] 

Willis's  Vowel  Resonances,  117c 

[Relative  force  of  the  partials  for  producing 

different  vowels,  j24f/] 
Relation    of    Strength    of    Resonance    to 
Alterations  of  Phase,  12oa 

Difference  of  pitch,  &c.,  necessary  to  reduce 
sympathetic  vibration  to  J^  of  that  pro- 
duced by  perfect  unisonance,  143a 

Numbers  from  wh  ich  fig.  52  was  constructed, 
145a 

Measurements  of  the  basilar  membrane  in 
a  new-born  child,  145c 

Alteration  of  size  of  Corti's  rods  as  they 
approach  the  vertex  of  the  cochlea,  145ci 

[Preyer's  distinguishable  and  undistin- 
guishable  intervals,  147f/] 

First  differential  tones  of  the  usual  har- 
monic intervals,  L'.4« 

[Differential  tones  of  different  orders  of  the 
usual  harmonic  intervals,  155(/] 

Different  intervals  which  would  give  33 
beats  of  their  primes,  172a 


[Pitch  numbers   of  Appunn's   bass   reeds, 

1776] 
[Experiments   on   audibility  of  very  deep 

tones,  177c] 
Coincident  partials  for  the  principal  con- 
sonances, 183a 
Pitch  numbers  of  the  primes  which  make 

consonant  iaitei-vals  with  a  tone  of  300 

vib.,  184c 
Beating  partials  of  the  notes  in  the  last 

table  with  a  note  of  301  vib.,  and  number 

of  beats,  184c^ 
Disturbance  of  a  consonance  by  altering 

one  of  its  tones  by  a  Semitone,  185c 
Influence  of  different  consonances  on  each 

other,  187b 
[Upper  partials  of  a  just  Fifth,  188d] 
[Upper  partials  of  an  altered  Fifth,  189c] 
[Comparison    of    the   upper    partials   of  a 

Fourth  and  Eleventh,  major  Sixth  and 

major    Thirteenth,     minor     Sixth     and 

minor  Thirteenth,  lS9c?and  1906,  c] 
[Comparison   of    the    upper   partials  of   a 

major  and  a  minor  Third,  190c?] 
[Comparison  of   the  uj^per  partials  of  aU 

the     usual    consonances,    pointing    out 

those  which  beat,  1916,  c] 
[Comparison     of     the    upper    partials     of 

septimal     consonances,     involving     the 

seventh  partial,  and  pointing  out  which 

beat,  195c,  d] 
[General  Table  of  the  first  16  harmonics  of 

C'66,  shewing  how  they  affect  each  other 

in  any  combination,  197c,  d] 
Table  of  partials  of  200  and  301,  shewing 

their  differential  tones,  198c 
Table  of  possible  triads,  shewing  consonant, 

dissonant,  and  septimal  intervals,  2126,  c 
Table  of  consonant  triads,  214a 
[The  first  16  harmonics  of  C,  2Ud] 
[Calculation  of  the  Combinational  Tones  of 

the  Major  Triads,  214rf] 
[Most  of  the  first  40  harmonics  oiA^,\f,  215c] 
[Calculation  of  the  Combinational  Tones  of 

the  Minor  Triads,  21:>d] 
[Calculation  of  the  Differential  Tones  of 

the  Major  Triads  in  their  most  Perfect 

Positions,  2l9d] 
[Calculation  of   the  Combinational  Tones 

of  the  Major  Triads  in  the  less  Perfect 

Positions,  220d] 
[Calculation  of  the  Combinational  Tones  of 


LIST  OF  TABLES. 


the  Minor  Triads  in  the  most  and  less 

Perfect   Positions  of   the  Minor  Triads, 

221d,  d'] 
[Calculation  of    the    false    Combinational 

Tones  of   Minor   Tetrads    in   their   best 

positions,  224f?] 
Ecclesiastical  Modes,  245c,  d 
Partial  Tones  of  the  Tonic,  257a 
[Pentatonic  Scales,  259c,  d] 
[Tetrachords    1  to    8,    with    intervals    in 

cents,  263d'] 
Greek  Diatonic  Scales,  267c 
[Greek  Diatonic  Scales  with  the  intervals 

in  cents,  268c] 
[Greek  Diatonic  Scales  reduced  to   begin- 
ning with  c,  with  the  intervals  in  cents, 

268f^'] 
Greek  modes  with  the  Greek  Ecclesiastical 

and  Helmholtzian  names,  269a 
Later  Greek  Scale,  270a 
Tonal  Keys,  270c 
Ecclesiastical  Scales  of  Ambrose  of  Milan, 

2716,  c 
The  Five  Melodic  Tonal  Modes,  272b 
[The    Seven   Ascending    and    Descending 

Scales,  compared  with  Greek,  with  inter- 
vals in  cents,  274e,  d] 
[The    different   scales    formed    by  a    dif- 
ferent choice  of   the    intercalary  tones, 

277c',  rf'] 
The  Five  Modes  with  variable  intercalary 

tones,  278a,  b 
[J.    Curwen's   characters   of  the  tones  in 

the  major  scale,  279&,  c] 
[Arabic   Scale  in   relation    to    the  major 

Thirds,  281rf'] 
Arabic  Scales,  2826-283c 
[Prof.   Land's  account   of    the  12    Arabic 

Scales,  284  note] 
Five  Modes  as  formed  from  three  chords 

each,  293c?,  294a 
The   same  with  double  intercalary  tones, 

297c,  d 
The  same,  final  form,  2986,  c 
Trichordal  Eelations  of  the  Tonal  Modes, 

:309rf 
[Thirds   and  Sixths   in   Just,    Equal,  and 

Pythagorean  Intonation  compared,  313c] 
[Combinational  Tones  of  Just,  Equal,  and 

Pythagorean  Intonation  compared,  314(i] 
The  Chordal  System  of  Prof.  Helmholtz's 

Just  Harmonium,  316c 
[Duodenary  statement  of  the  tones  on  Prof. 

Helmholtz's  Just  Harmonium,  317c,  d] 
The   Chordal   System   of   the  minor   keys 

on  Prof.  Helmholtz's  Just  Harmonium, 

318a,  b,  d 
[Table  of  the  relation  of  the  Cycle  of  58  to 

Just  Intonation,  3296,  c] 
[Tabular  Expression  of   the  Diagram,  fig. 

61,  332] 
[Table  of  Roughness,  3ZM] 


Measurements  of  Glass  Resonators,  373c 

Measurements  of  resonance  tubes  men- 
tioned on  p.  55a,  Z77d 

Table  of  tones  of  a  conical  pipe  of  zinc, 
calculated  from  formula  393c  [with  sub- 
sidiary tables,  393c?  and  394c] 

Table  of  Mayer's  observations  on  numbers 
of  beats,  418a 

Table  of  four  stops  for  a  single  manual 
justly  intoned  instrument,  421c 

Table  of  five  stops  for  the  same,  422a 

In  the  Additions  by  Translator. 
Table  of  Pythagorean  Intonation,  4336,  c 
Table  of  Meantone  Intonation,  4346 
Table  of  Equal  Intonation,  437c,  d 
Synonymity  of  Equal  Temperament,  4386 
Synonymity  of  Mr.  Bosanquet's  Notes  in 

Fifths,  439a 
Notes  of  Mr.  Bosanquet's  Cycle  of  53  in 

order  of  Pitch,  4396,  c,  d 
Expression  of  Just  Intonation  in  the  Cycle 

of  1200,  p.  440 
Principal   Table   for  calculation  of  cents, 

450a,  Auxiliary  Tables,  451a 
Table  of  Intervals  not  exceeding  one  Octave, 

4536 
Unevenly  numbered  Harmonics  up  to  the 

D3rd,  457a 
Number  of  any  Interval  not  exceeding  a 

Tritone,  contained  in  an  Octave,  457c 
Harmonic  Duodene  or  Unit  of  Modulation, 

461a 
The  Duodenarium,  463a 
Fingerboard  of  the  Harmonical,  first  four 

Octaves,  with  scheme,  4676,  fifth  Octave, 

468rf 
Just  Harmonium  scheme,  470a 
Just  English  Concertina  scheme,  4706 
Mr.    Colin    Brown's    Voice    Harmonium 

Fingerboard  and  scheme,  471a 
Rev.   Henry  Liston's  Organ  and  scheme, 

4736 
Gen.  Perronet  Thompson's  Organ  scheme, 

A7Zd 
Mr.  H.  Ward  Poole's  100  tones,  474c 
Mr.  H.  W.  Poole's  scheme  for  keys  of  F, 

C,  G,  476a 
Mv.    Bosanquet's    Generalised    Keyboard, 

480 
Expression  of  the  degrees  of  the  53  divi- 
sion by  multiples  of  2,  5  and  7,  p.  481c 
Typographical  Plan  of  Mr.  J.  Paul  White's 

Fingerboard,  4826 
Specimens   of   tuning  in  Meantone  Tem- 
perament, 484c 
Specimens  of  tuning  in  Equal  Tempera- 
ment, 4856 
Pianoforte   Tuning — Fourths   and   Fifths, 

485d 
Cornu    and    Mercadier's    observation    on 

Violin  Intonation,  486c  to  4876 


LIST  OF  TABLES. 


Scheme  for  tuning  in  Equal  Temperament, 
4895 

Proof  of  rule  for  tuning  in  Equal  Tempera- 
ment, 490e,  d 

Proof  of  rule  for  Tuning  in  Meantoue  Tem- 
perament, 492« 

Historical  Pitches  in  order  from  Lowest  to 
Highest,  49  5«  to  504rt 

Classified  Index  to  the  last  Table,  ri04&  to 

Effects  of  the  length  of  the  foot  in  differ- 
ent countries  on  the  pitch  of  organs, 
512a 

Non-harmonic  scales,  514c  to  519c 


Vowel  sound  '  Oh  ! '      Analysis  at  various 

pitches  by  Messrs.  Jenkin  &  Ewing,  539d 

to  5416 
Vowel   sounds  '  oo,'   'awe,'   'ah,'  analysis 

at   various    pitches    by   Messrs.   Jenkin 

&  Ewing,  p.  541c,  d 
Mean  and  actual  Compass  of  the  Human 

Voice,  545«,  b,  e 
True   Tritonic,    False   Tritonic,    Zarlino's, 

Meantone    and    Equal    Temperaments, 

compared,  548a  

Presumed  Characters  of  Major  and  Minor 

Keys,  551-«,  h 


INTEODUCTION. 


In  the  present  work  an  attempt  will  be  made  to  connect  the  boundaries 
of  two  sciences,  which,  although  drawn  towards  each  other  by  many 
natural  atiinities,  have  hitherto  remained  practically  distinct — I  mean  the 
boundaries  of  physical  and  physiological  acoustics  on  the  one  side,  and  of 
musical  science  and  esthetics  on  the  other.  The  class  of  readers  addressed 
will,  consequently,  have  had  very  different  cultivation,  and  will  be  affected 
by  very  different  interests.  It  will  therefore  not  be  superfluous  for  the 
author  at  the  outset  distinctly  to  state  his  intention  in  undertaking  the 
work,  and  the  aim  he  has  sought  to  attain.  The  horizons  of  physics, 
philosophy,  and  art  have  of  late  been  too  widely  separated,  and,  as  a 
consequence,  the  language,  the  methods,  and  the  aims  of  any  one  of  these 
studies  present  a  certain  amount  of  difficulty  for  the  student  of  any  other  H 
of  them ;  and  possibly  this  is  the  principal  cause  why  the  problem  here 
undertaken  has  not  been  long  ago  more  thoroughly  considered  and  advanced 
towards  its  solution. 

It  is  true  that  acoustics  constantly  employs  conceptions  and  names 
borrowed  from  the  theory  of  harmony,  and  speaks  of  the  'scale,'  'intervals,' 
'  consonances,'  and  so  forth ;  and  similarly,  manuals  of  Thorough  Bass 
generally  begin  with  a  physical  chapter  which  speaks  of  '  the  numbers  of 
vibrations,'  and  fixes  their  'ratios'  for  the  different  intervals;  but,  up  to 
the  present  time,  this  apparent  connection  of  acoustics  and  music  has  been 
wholly  external,  and  may  be  regarded  rather  as  an  expression  given  to  the 
feelmg  that  such  a  connection  must  exist,  than  as  its  actual  formulation. 
Physical  knowledge  may  indeed  have  been  useful  for  musical  instrument 
makers,  but  for  the  development  and  foundation  of  the  theory  of  harmony  H 
It  has  hitherto  been  totally  barren.  And  yet  the  essential  facts  within  the 
field  here  to  be  explained  and  turned  to  account,  have  been  known  from  the 
earliest  times.  Even  Pythagoras  (fl.  circa  B.C.  540-510)  knew  that  when 
strings  of  different  lengths  but  of  the  same  make,  and  subjected  to  the 
same  tension,  were  used  to  give  the  perfect  consonances  of  the  Octave, 
Fifth,  or  Fourth,  their  lengths  must  be  in  the  ratios  of  1  to  2,  2  to  '6,  or 
3  to  4  respectively,  and  if,  as  is  probable,  his  knowledge  was  partly  derived 
from  the  Egyptian  priests,  it  is  impossible  to  conjecture  in  what  remote 
antiquity  this  law  was  first  known.  Later  physics  has  extended  the  law  of 
Pythagoras  by  passing  from  the  lengths  of  strings  to  the  number  of  vibra- 
tions, and  thus  making  it  applicable  to  the  tones  of  all  musical  instruments, 

and  the  numerical  relations  4  to  5  and  5  to  «i  have  been  added  to  the  above 

u 


-  PLAN  OF  THE  WORK.  introd. 

for  the  less  perfect  consonances  of  the  major  and  minor  Thirds,  but  I  am 
not  aware  that  any  real  step  was  ever  inade  towards  answering  the  ques- 
tion :  What  have  musical  consonances  to  do  ivith  the  ratios  of  the  first  six 
numbers  !  Musicians,  as  well  as  philosophers  and  physicists,  have  generally 
contented  themselves  with  saying  in  effect  that  human  minds  were  in  some 
unknown  manner  so  constituted  as  to  discover  the  numerical  relations  of 
musical  vibrations,  and  to  have  a  peculiar  pleasure  in  contemplating  simple 
ratios  which  are  readily  comprehensible. 

Meanwhile  musical  esthetics  has  made  unmistakable  advances  in  those 
points  which  depend  for  their  solution  rather  on  psychological  feeling  than 
on  the  action  of  the  senses,  by  introducing  the  conception  of  movement  in 

IT  the  examination  of  musical  works  of  art.  E.  Hanslick,  in  his  book  On  the 
Beautiful  in  Music  {Ueher  das  musihalisch  Schone),  triumphantly  attacked 
the  false  standpoint  of  exaggerated  sentimentality,  from  which  it  was 
fashionable  to  theorise  on  music,  and  referred  the  critic  to  the  simple 
elements  of  melodic  movement.  The  esthetic  relations  for  the  structure  of 
musical  compositions,  and  the  characteristic  differences  of  individual  forms 
of  composition  are  explained  more  fully  in  Vischer's  Esthetics  (Aesthetik). 
In  the  inorganic  world  the  kind  of  motion  we  see,  reveals  the  kind  of  moving 
force  in  action,  and  in  the  last  resort  the  only  method  of  recognising  and 
measuring  the  elementary  powers  of  nature  consists  in  determining  the 
motions  they  generate,  and  this  is  also  the  case  for  the  motions  of  bodies 
or  of  voices  which  take  place  under  the  influence  of  human  feelings.    Hence 

^the  properties  of  musical  movements  which  possess  a  graceful,  dallying,  or 
a  heavy,  forced,  a  dull,  or  a  powerful,  a  quiet,  or  excited  character,  and  so 
on,  evidently  chiefly  depend  on  psychological  action.  In  the  same  way 
questions  relating  to  the  equilibrium  of  the  separate  parts  of  a  musical 
composition,  to  their  development  from  one  another  and  their  connection 
as  one  clearly  intelligible  whole,  bear  a  close  analogy  to  similar  questions 
in  architecture.  But  all  such  investigations,  however  fertile  they  may  have 
been,  cannot  have  been  otherwise  than  imperfect  and  uncertain,  so  long  as 
they  were  without  their  proper  origin  and  foundation,  that  is,  so  long  as 
there  was  no  scientific  foundation  for  their  elementary  rules  relating  to  the 
construction  of  scales,  chords,  keys  and  modes,  in  short,  to  all  that  is 
usually  contained  in  works  on  '  Thorough  Bass '.    In  this  elementary  region 

U  we  have  to  deal  not  merely  with  unfettered  artistic  inventions,  but  with  the 
natural  power  of  immediate  sensation.  Music  stands  in  a  much  closer 
connection  with  pure  sensation  than  any  of  the  other  arts.  The  latter 
rather  deal  with  what  the  senses  apprehend,  that  is  with  the  images  of 
outward  objects,  collected  by  psychical  processes  from  immediate  sensation. 
Poetry  aims  most  distinctly  of  all  at  merely  exciting  the  formation  of 
images,  by  addressing  itself  especially  to  iinagination  and  memory,  and  it 
is  only  by  subordinate  auxiliaries  of  a  more  musical  kind,  such  as  rhythm, 
and  imitations  of  sounds,  that  it  appeals  to  the  immediate  sensation  of 
hearing.  Hence  its  efltects  depend  mainly  on  psychical  action.  The  plastic 
arts,  although  they  make  use  of  the  sensation  of  sight,  address  the  eye 
almost  in  the  same  way  as  poetry  addresses  the  ear.  Their  main  purpose 
is  to  excite  in  us  the  image  of  an  external  object  of  determinate  form  and 
colour.     The  spectator  is  essentially  intended  to  interest  himself  in  this 


iN-TROD.  PLAN  OF  THE  WORK.  3 

image,  and  enjoy  its  beauty  ;  not  to  dwell  upon  the  means  by  which  it  was 
created.  It  must  at  least  be  allowed  that  the  pleasure  of  a  connoisseur  or 
virtuoso  in  the  constructive  art  shown  in  a  statue  or  a  picture,  is  not  an 
essential  element  of  artistic  enjoyment. 

It  is  only  in  painting  that  we  find  colour  as  an  element  which  is  directly 
appreciated  by  sensation,  without  any  intervening  act  of  the  intellect.  On 
the  contrary,  in  music,  the  sensations  of  tone  are  the  material  of  the  art. 
So  far  as  these  sensations  are  excited  in  music,  we  do  not  create  out  of 
them  any  images  of  external  objects  or  actions.  Again,  when  in  hearing  a 
concert  we  recognise  one  tone  as  due  to  a  violin  and  another  to  a  clarinet, 
our  artistic  enjoyment  does  not  depend  upon  our  conception  of  a  violin  or 
clarinet,  but  solely  on  our  hearing  of  the  tones  they  produce,  whereas  the  ^ 
artistic  enjoyment  resulting  from  viewing  a  marble  statue  does  not  depend 
on  the  white  light  which  it  reflects  into  the  eye,  but  upon  the  mental  image 
of  the  beautiful  human  form  which  it  calls  up.  In  this  sense  it  is  clear 
that  music  has  a  more  immediate  connection  with  pure  sensation  than  any 
other  of  the  fine  arts,  and,  consequentl}^,  that  the  theory  of  the  sensations 
of  hearing  is  destined  to  play  a  much  more  important  part  in  musical 
esthetics,  than,  for  example,  the  theory  of  chiaroscuro  or  of  perspective  in 
painting.  Those  theories  are  certainly  useful  to  the  artist,  as  means  for 
attaining  the  most  perfect  representation  of  nature,  but  they  have  no  part 
in  the  artistic  effect  of  his  work.  In  music,  on  the  other  hand,  no  such 
perfect  representation  of  nature  is  aimed  at ;  tones  and  the  sensations  of 
tone  exist  for  themselves  alone,  and  produce  their  effects  independently  "^ 
of  anything  behind  them. 

This  theory  of  the  sensations  of  hearing  belongs  to  natural  science,  and 
comes  in  the  first  place  under  ^A?/sio/o^/<;«/ rtco/^s^/c.s\^  Hitherto  it  is  the 
physical  part  of  the  theory  of  sound  that  has  been  almost  exclusively  treated 
at  length,  that  is,  the  investigations  refer  exclusively  to  the  motions  produced 
by  solid,  liquid,  or  gaseous  bodies  when  they  occasion  the  sounds  which  the 
ear  appreciates.  This  physical  acoustics  is  essentially  nothing  but  a  section 
of  the  theory  of  the  motions  of  elastic  bodies.  It  is  physically  indifferent 
whether  observations  are  made  on  stretched  strings,  by  means  of  spirals  of 
brass  wire  (which  vibrate  so  slowly  that  the  eye  can  easily  follow  their 
motions,  and,  consequently,  do  not  excite  any  sensation  of  sound),  or  by 
means  of  a  violin  string  (where  the  eye  can  scarcely  perceive  the  vibrations  ^i 
which  the  ear  readily  appreciates).  The  laws  of  vibratory  motion  are  pre- 
cisely the  same  in  both  cases ;  its  rapidity  or  slowness  does  not  affect  the 
laws  themselves  in  the  slightest  degree,  although  it  compels  the  observer  to 
apply  different  methods  of  observation,  the  eye  for  one  and  the  ear  for 
the  other.  In  physical  acoustics,  therefore,  the  phenomena  of  hearing  are 
taken  into  consideration  solely  because  the  ear  is  the  most  convenient  and 
handy  means  of  observing  the  more  rapid  elastic  vibrations,  and  the  physicist 
is  compelled  to  study  the  peculiarities  of  the  natural  instrument  which  he  is 
employing,  in  order  to  control  the  correctness  of  its  indications.  In  this 
way,  although  physical  acoustics  as  hitherto  pursued,  has,  undoubtedly, 
collected  many  observations  and  much  knowledge  concerning  the  action  of 
the  ear,  which,  therefore,  belong  to physiolocjical  aconstics,  these  results  were 
not  the  principal  object  of  its  investigations  ;  they  were  merely  secondary 

B  2 


4  PLAN  OF  THE  WORK.  introd. 

and  isolated  facts.  The  only  justification  for  devoting  a  separate  chapter 
to  acoustics  in  the  theory  of  the  motions  of  elastic  bodies,  to  which  it 
essentially  belongs,  is,  that  the  application  of  the  ear  as  an  instrument 
of  research  influenced  the  nature  of  the  experiments  and  the  methods  of 
observation. 

But  in  addition  to  a  physical  there  is  a  physiological  theory  of  acousticSy 
the  aim  of  v^hich  is  to  investigate  the  processes  that  take  place  within  the 
ear  itself.  The  section  of  this  science  which  treats  of  the  conduction  of  the 
motions  to  which  sound  is  due,  from  the  entrance  of  the  external  ear  to  the 
expansions  of  the  nerves  in  the  labyrinth  of  the  inner  ear,  has  received 
much  attention,  especially  in  Germany,  since  ground  was  broken  by 
11  Johannes  Mueller.  At  the  same  time  it  must  be  confessed  that  not  many 
results  have  as  yet  been  established  with  certainty.  But  these  attempts 
attacked  only  a  portion  of  the  problem,  and  left  the  rest  untouched. 
Investigations  into  the  processes  of  each  of  our  organs  of  sense,  have  in 
general  three  different  parts.  First  we  have  to  discover  how  the  agent 
reaches  the  nerves  to  be  excited,  as  light  for  the  eye  and  sound  for  the  ear. 
This  may  be  called  the  physical  part  of  the  corresponding  physiological 
investigation.  Secondly  we  have  to  investigate  the  various  modes  in  which 
the  nerves  themselves  are  excited,  giving  rise  to  their  various  sensations, 
and  finally  the  laws  according  to  which  these  sensations  result  in  mental 
images  of  determinate  external  objects,  that  is,  in  perceptions.  Hence  we 
have  secondly  a  specially  physiological  investigation  for  sensations,  and 
11  thirdly,  a  specially  psychological  investigation  for  perceptions.  Now  whilst 
the  physical  side  of  the  theory  of  hearing  has  been  already  frequently 
attacked,  the  results  obtained  for  its  physiological  and  psychological 
sections  are  few,  imperfect,  and  accidental.  Yet  it  is  precisely  the  physio- 
logical part  in  especial — the  theory  of  the  sensations  of  hearing — to  which 
the  theory  of  music  has  to  look  for  the  foundation  of  its  structure. 

In  the  present  work,  then,  I  have  endeavoured  in  the  first  place  to  collect 
and  arrange  such  materials  for  the  theory  of  the  sensations  of  hearing  as 
already  existed,  or  as  I  was  able  to  add  from  my  own  personal  investigations. 
Of  course  such  a  first  attempt  must  necessarily  be  somewhat  imperfect,  and 
be  limited  to  the  elements  and  the  most  interesting  divisions  of  the  subject 
discussed.  It  is  in  this  light  that  I  wish  these  studies  to  be  regarded. 
11  Although  in  the  propositions  thus  collected  there  is  little  of  entn-ely  new 
discoveries,  and  although  even  such  apparently  new  facts  and  observations 
as  they  contain  are,  for  the  most  part,  more  properly  speaking  the  imme- 
diate consequences  of  my  having  more  completely  carried  out  known 
theories  and  methods  of  investigation  to  their  legitimate  consequences, 
and  of  my  having  more  thoroughly  exhausted  their  results  than  had  hare- 
tofore  been  attempted,  yet  I  cannot  but  think  that  the  facts  frequently 
receive  new  importance  and  new  illumination,  by  being  regarded  from  a 
fresh  point  of  view  and  in  a  fresh  connection. 

The  First  Part  of  the  following  investigation  is  essentially  physical  and 
physiological.  It  contains  a  general  investigation  of  the  phenomenon  of 
harmonic  uppier partial  tones.  The  nature  of  this  phenomenon  is  established, 
and  its  relation  to  qnality  of  tone  is  proved.  A  series  of  qualities  of  tone  are 
analysed  in  respect  to  their  harmonic  upper  partial  tones,  and  it  results 


iNTROD.  PLAN    OF   THE   WORK.  5 

that  these  upper  partial  tones  are  not,  as  was  hitherto  thought,  isolated 
phenomena  of  small  importance,  but  that,  with  very  few  exceptions,  they 
determine  the  qualities  of  tone  of  almost  all  instruments,  and  are  of  the 
greatest  importance  for  those  qualities  of  tone  which  are  best  adapted  for 
musical  purposes.  The  question  of  how  the  ear  is  able  to  perceive  these 
harmonic  upper  partial  tones  then  leads  to  an  hypothesis  respecting  the 
mode  in  which  the  auditory  nerves  are  excited,  which  is  well  fitted  to 
reduce  all  the  facts  and  laws  in  this  department  to  a  relatively  simple 
mechanical  conception. 

The  Second  Part  treats  of  the  disturbances  produced  by  the  simultaneous 
production  of  two  tones,  namely  the  comhimitional  tones  and  heat:.  The 
physiologico-physical  investigation  shows  that  two  tones  can  besimul-^ 
taneously  heard  by  the  ear  without  mutual  disturbance,  when  and  only 
when  they  stand  to  each  other  in  the  perfectly  determinate  and  well-known 
relations  of  intervals  which  form  musical  consonance.  We  are  thus  imme- 
diately introduced  into  the  field  of  music  proper,  and  are  led  to  discover 
the  physiological  reason  for  that  enigmatical  numerical  relation  announced 
by  Pythagoras.  The  magnitude  of  the  consonant  intervals  is  independent 
of  the  quality  of  tone,  but  the  harmoniousness  of  the  consonances,  and  the 
distinctness  of  their  separation  from  dissonances,  depend  on  the  quality  of 
tone.  The  conclusions  of  physiological  theory  here  agree  precisely  with  the 
musical  rules  for  the  formation  of  chords  ;  they  even  go  more  into  par- 
ticulars than  it  was  possible  for  the  latter  to  do,  and  have,  as  I  believe,  the 
authority  of  the  best  composers  in  their  favour.  ^ 

In  these  first  two  Parts  of  the  book,  no  attention  is  paid  to  esthetic 
considerations.  Natural  phenomena  obeying  a  blind  necessity,  are  alone 
treated.  The  Third  Part  treats  of  the  construction  of  musical  scales  and 
)wtes.  Here  we  come  at  once  upon  esthetic  ground,  and  the  differences  of 
national  and  individual  tastes  begin  to  appear.  Modern  music  has  especially 
developed  the  principle  of  tonality,  which  connects  all  the  tones  in  a  piece 
of  music  by  their  relationship  to  one  chief  tone,  called  the  tonic.  On 
admitting  this  principle,  the  results  of  the  preceding  investigations  furnish 
a  method  of  constructing  our  modern  musical  scales  and  modes,  from 
which  all  arbitrary  assumption  is  excluded. 

I  was  unwilling  to  separate  the  physiological  investigation  from  its 
musical  consequences,  because  the  correctness  of  these  consequences  must  H 
be  to  the  physiologist  a  verification  of  the  correctness  of  the  physical  and 
physiological  views  advanced,  and  the  reader,  who  takes  up  my  book  for  its 
musical  conclusions  alone,  cannot  form  a  perfectly  clear  view  of  the  meaning 
and  bearing  of  these  consequences,  unless  he  has  endeavoured  to  get  at 
least  some  conception  of  their  foundations  in  natural  science.  But  in 
order  to  facihtate  the  use  of  the  book  by  readers  w^ho  have  no  special 
knowledge  of  physics  and  mathematics,  I  have  transferred  to  appendices, 
at  the  end  of  the  book,  all  special  instructions  for  performing  the  more 
comphcated  experiments,  and  also  all  mathematical  investigations.  These 
appendices  are  therefore  especially  intended  for  the  physicist,  and  contain 
the  proofs  of  my  assertions.*  In  this  way  I  hope  to  have  consulted  the 
interests  of  both  classes  of  readers. 

*  [The  additional  Appendix  XX.  bj'  the  Translator  is  intended  especially  for  the  use  of 
musical  students. — Translator.] 


6  PLAN    OF   THE    WORK.  ixtrod. 

It  is  of  course  impossible  for  any  one  to  understand  the  investigations 
thoroughly,  who  does  not  take  the  trouble  of  becoming  acquainted  by  per- 
sonal observation  with  at  least  the  fundamental  phenomena  mentioned. 
Fortunately  with  the  assistance  of  common  musical  instruments  it  is  easy 
for  any  one  to  become  acquainted  with  harmonic  upper  partial  tones,  com- 
binational tones,  beats,  and  the  like.*  Personal  observation  is  better  than 
the  exactest  description,  especially  when,  as  here,  the  subject  of  investiga- 
tion is  an  analysis  of  sensations  themselves,  which  are  always  extremely 
difficult  to  describe  to  those  who  have  not  experienced  them. 

In  my  somewhat  unusual  attempt  to  pass  from  natural  philosophy  into 
the  theory  of  the  arts,  I  hope  that  I  have  kept  the  regions  of  physiology 

H  and  esthetics  sufficiently  distinct.  But  I  can  scarcely  disguise  from  myself, 
that  although  my  researches  are  confined  to  the' lowest  grade  of  musical 
grammar,  they  may  probably  appear  too  mechanical  and  unworthy  of  the 
dignity  of  art,  to  those  theoreticians  who  are  accustomed  to  summon  the 
enthusiastic  feelings  called  forth  by  the  highest  works  of  art  to  the  scientific 
investigation  of  its  basis.  To  these  I  would  simply  remark  in  conclusion, 
that  the  following  investigation  really  deals  only  with  the  analysis  of 
actually  existing  sensations — that  the  physical  methods  of  observation 
employed  are  almost  solely  meant  to  facilitate  and  assure  the  work  of  this 
analysis  and  check  its  completeness — and  that  this  analysis  of  the  sensations 
would  suffice  to  furnish  all  the  results  required  for  musical  theory,  even 
independently  of  my  physiological  hypothesis  concerning  the  mechanism  of 

^  hearing,  already  mentioned  (p.  oa),  but  that  I  was  unwilling  to  omit  that 
hypothesis  because  it  is  so  well  suited  to  furnish  an  extremely  simple  con- 
nection between  all  the  very  various  and  very  complicated  phenomena 
which  present  themselves  in  the  course  of  this  investigation. t 

*  [But  the  use  of  the  H(trmonical,  described  London,  ]Macmillan,  1873.     Such  readers  will 

in  App.  XX.  sect.  F.  No.  1,  and  invented  for  also  find   a  clear  exposition  of  the  physical 

the  purpose  of  illustrating  the  theories  of  this  relations  of  sound  in  J.  Tyndall,  On  Souiul, 

work,  is  recommended  as  greatly  superior  for  a  course  of  eight  lectures,  London,   1867,  (the 

students  and  teachers  to  any  other  instrument.  last  or  fourth  edition  188.3)  Longmans,  Green, 

—  Transhitor.']  &   Co.      A  German  translation  of  this  work, 

t  Readers  unaccustomed  to  mathematical  entitled  Der  Schall,  edited  by  H.  Helmholtz 

and    physical     considerations    will     find    an  and  G.  Wiedemann,  was  published  at  Bruns- 

abridged  account  of  the  essential  contents  of  wick  in  1874. 
this  Ijook  in  Sedley  Taylor,  Sound  and  Musk, 


*^*  [The  marks  ^  in  the  outer  margin  of  each  page,  separate  the  page  into 
4  sections,  referred  to  as  a,  >>,  c,  d,  placed  after  the  number  of  the  page.  If  any 
section  is  in  doul)le  columns,  the  letter  of  the  second  column  is  accented,  as 
p.  i:3.r.] 


PART     I. 

ON    THE    COMPOSri'ION    OF    YIBHATIONS, 


uppp:r  partial  tones,  a^td  qualttib:s  of  toxe. 

CHAPTER   I. 

ox    THE    SENSATION    OF    SOUND    IN    GENERAL. 

Sensations  result  from  the  action  of  an  external  stimulus  on  the  sensitive  apparatus 
of  oiir  nerves.  Sensations  differ  in  kind,  partly  with  the  organ  of  sense  excited, 
and  partly  with  the  nature  of  the  stimulus  employed.  Each  organ  of  sense  pro- 
duces peculiar  sensations,  which  cannot  be  excited  by  means  of  any  other;  the 
eye  gives  sensations  of  light,  the  ear  sensations  of  sound,  the  skin  sensations  of 
touch.  Even  when  the  same  sunbeams  which  excite  in  the  eye  sensations  of  light, 
impinge  on  the  skin  and  excite  its  nerves,  they  are  felt  only  as  heat,  not  as  light,  wt 
In  the  same  way  the  vibration  of  elastic  bodies  heard  by  the  ear,  can  also  be  felt 
by  the  skin,  but  in  that  case  produce  only  a  whirring  fluttering  sensation,  not 
sound.  The  sensation  of  sound  is  therefore  a  species  of  reaction  against  external 
stimulus,  peculiar  to  the  ear,  and  excitable  in  no  other  organ  of  the  body,  and  is 
completely  distinct  from  the  sensation  of  any  other  sense. 

As  our  problem  is  to  study  the  laws  of  the  sensation  of  hearing,  our  fir:jt 
business  will  be  to  examine  how  many  kinds  of  sensation  the  ear  can  generate,  and 
what  differences  in  the  external  means  of  excitement  or  sound,  correspond  to  these 
differences  of  sensation. 

The  first  and  principal  difference  between  various  sounds  experienced  by  our  ear, 
is  that  between  7ioises  and  musical  tom^s.  The  soughing,  howling,  and  whistling 
of  the  wind,  the  splashing  of  water,  the  rolling  and  rumbling  of  carriages,  are 
examples  of  the  first  kind,  and  the  tones  of  all  musical  instruments  of  the  second. 
Noises  and  musical  tones  may  certainly  intermingle  in  very  various  degrees,  and  *t 
pass  insensibly  into  one  another,  but  their  extremes  are  widely  separated. 

The  nature  of  the  difference  between  musical  tones  and  noises,  can  generally 
be  determined  by  attentive  aural  observation  without  artificial  assistance.  We 
perceive  that  generally,  a  noise  is  accompanied  by  a  rapid  alternation  of  different 
kinds  of  sensations  of  sound.  Think,  for  example,  of  the  rattling  of  a  carriage 
over  granite  paving  stones,  the  splashing  or  seething  of  a  waterfall  or  of  the  waves 
of  the  sea,  the  rustling  of  leaves  in  a  wood.  In  all  these  cases  we  have  rapid, 
irregular,  but  distinctly  perceptible  alternations  of  vr.rious  kinds  of  sounds,  which 
crop  up  fitfully.  When  the  wind  howls  the  alternation  is  slow,  the  sound  slowly 
and  gradually  rises  and  then  falls  again.  It  is  also  more  or  less  possible  to  separate 
restlessly  alternating  sounds  in  case  of  the  greater  number  of  other  noises.  We 
shall  hereafter  become  acquainted  with  an  instrument,  called  a  resonator,  which 
will  materially  assist  the  ear  in  making  this  separation.  On  the  other  hand,  a 
musical   tone   strikes  the   ear  as    a   perfectly  luidisturbed,    luiiform    sound   which 


8  NOISE  AND  MUSICAL  TONE.  tart  i. 

remains  unaltered  as  long  as  it  exists,  and  it  presents  no  alternation  of  various 
kinds  of  constituents.  To  this  then  corresponds  a  simple,  regular  kind  of  sensation, 
whereas  in  a  noise  many  various  sensations  of  musical  tone  are  irregularly  mixed 
up  and  as  it  were  tumbled  about  in  confusion.  We  can  easily  compound  noises 
out  of  musical  tones,  as,  for  example,  by  simultaneously  striking  all  the  keys  con- 
tained in  one  or  two  octaves  of  a  pianoforte.  This  shows  us  that  musical  tones 
are  the  simpler  and  more  regular  elements  of  the  sensations  of  hearing,  and  that 
we  have  consequently  first  to  study  the  laws  and  peculiarities  of  this  class  of 
sensations. 

Then  comes  the  further  question  :  On  what  difterence  in  the  external  means  of 
excitement  does  the  difference  between  noise  and  musical  tone  depend  1  The 
normal  and  usual  means  of  excitement  for  the  human  ear  is  atmospheric  vibration. 
^  Tiie  irregularly  alternating  sensation  of  the  ear  in  the  case  of  noises  leads  us  to 
conclude  that  for  these  the  vibi-ation  of  the  air  must  also  change  irregularl}'.  For 
musical  tones  on  the  other  hand  we  anticipate  a  regular  motion  of  the  air,  con- 
tiniiing  uniformly,  and  in  its  turn  excited  by  an  equally  regular  motion  of  the 
sonorous  body,  whose  impulses  were  conducted  to  the  ear  by  the  air. 

Those  regular  motions  which  produce  musical  tones  have  been  exactly  investi- 
gated by  physicists.  They  are  oscillations,  vibrations,  or  swings,  that  is,  up  and 
down,  or  to  and  fro  motions  of  sonorous  bodies,  and  it  is  necessary  that  these 
oscillations  should  be  regularly  perioilic.  By  a  periodic  motion  we  mean  one  which 
constantly  returns  to  the  same  condition  after  exactly  equal  intervals  of  time.  The 
length  of  the  equal  intervals  of  time  between  one  state  of  the  motion  and  its  next 
exact  repetition,  we  call  the  length  of  the  oscillation,  vibration,  or  swing,  or  the 
period  of  the  motion.  In  what  manner  the  moving  body  actually  moves  during 
one  period,  is  perfectly  inditterent.  As  illustrations  of  periodical  motion,  take  the 
^motion  of  a  clock  pendulum,  of  a  stone  attached  to  a  string  and  whirled  round  in 
a  circle  with  uniform  velocity,  of  a  hammer  made  to  rise  and  fall  uniformly  by  its 
connection  with  a  water  wheel.  All  these  motions,  however  different  be  their 
form,  are  periodic  in  the  sense  here  explained.  The  length  of  their  periods,  which 
in  the  cases  adduced  is  generally  from  one  to  several  seconds,  is  relatively  long  in 
comparison  with  the  much  shorter  periods  of  the  vibrations  producing  nuisical 
tones,  the  lowest  or  deepest  of  which  makes  at  least  30  in  a  second,  while  in  other 
cases  their  number  may  increase  to  several  thousand  in  a  second. 

Our  definition  of  periodic  motion  then  enables  us  to  answer  the  question  pro- 
posed as  follows  : — The  sensation  of  a  musical  tone  is  due  to  a  rapid  periodic 
inotion  of  the  sonorous  body ;    the  sensation  of  a  noise  to  non-periodic  motions. 

The  musical  vibrations  of  solid  bodies  are  often  visible.  Although  they  may 
be  too  rapid  for  the  eye  to  follow  them  singly,  wc  easily  recognise  that  a  sounding- 
string,  or  tuning-fork,  or  the  tongue  of  a  reed-pipe,  is  rapidly  vibrating  between  two 
^  fixed  limits,  and  the  regulai",  apparently  immovable  image  that  we  see,  notwith- 
standing the  real  motion  of  the  body,  leads  us  to  conclude  that  the  backward  and 
forward  motions  are  quite  regular.  In  other  cases  we  can  feel  the  swinging  motions 
of  sonorous  solids.  Thus,  the  player  feels  the  trembling  of  the  reed  in  the  mouth- 
piece of  a  clarinet,  oboe,  or  bassoon,  or  of  his  own  lips  in  the  mouthpieces  of 
trumpets  and  trombones. 

The  motions  proceeding  from  the  sounding  bodies  are  usually  conducted  to  our 
ear  by  means  of  the  atmosphere.  The  particles  of  air  must  also  execute  periodi- 
cally recurrent  vibrations,  in  order  to  excite  the  sensation  of  a  musical  tone  in  our 
ear.  This  is  actually  the  case,  although  in  daily  experience  sound  at  first  seems 
to  be  some  agent,  which  is  constantly  advancing  through  the  air,  and  projjagating 
itself  further  and  further.  We  must,  however,  here  distinguish  between  the  motion 
of  the  individual  particles  of  air — which  takes  place  periodically  backwards  and 
forwards  within  very  narrow  limits — and  the  propagation  of  the  sonorous  tremor. 
The  latter  is  constantly  advancing  by  the  constant  attraction  of  fresh  particles  into 
its  sphere  of  tremor. 


CHAP.  I.  PROPAGATION  OF  SOUND.  9 

This  is  a  peculiarity  of  all  so-called  ii)i<hd(tto)-r/  motions.  Suppose  a  stone  to 
be  thrown  into  a  piece  of  calm  water.  Round  the  spot  struck  there  forms  a  little 
ring  of  wave,  which,  advancing  equally  in  all  directions,  expands  to  a  constantly 
increasing  circle.  Corresponding  to  this  ring  of  wave,  sound  also  proceeds  in  the 
air  from  the  excited  point  and  advances  in  all  directions  as  far  as  the  limits  of  the 
mass  of  air  extend.  The  process  in  the  air  is  essentially  identical  with  that  on  the 
surface  of  the  water.  The  principal  difference  consists  in  the  spherical  propagation 
of  sound  in  all  directions  through  the  atmosphere  which  fills  all  surrounding  space, 
whereas  the  waves  of  the  water  can  only  advance  in  rings  or  circles  on  its  surface. 
The  crests  of  the  waves  of  water  correspond  in  the  waves  of  sound  to  spherical 
shells  where  the  air  is  condensed,  and  the  troughs  to  shells  where  it  is  rarefied. 
On  the  free  surface  of  the  water,  the  mass  when  compressed  can  slip  upwards  and 
so  form  ridges,  but  in  the  interior  of  the  sea  of  air,  the  mass  must  be  condensed, 
as  there  is  no  unoccupied  spot  for  its  escape.  «[I 

The  waves  of  water,  therefore,  continually  advance  without  returning.  But 
we  nuist  not  suppose  that  the  particles  of  water  of  which  the  waves  are  composed 
advance  in  a  similar  manner  to  the  waves  themselves.  The  motion  of  the  particles 
of  water  on  the  surface  can  easily  be  rendered  visible  b}^  floating  a  chip  of  wood 
upon  it.  This  will  exactly  share  the  motion  of  the  adjacent  particles.  Now,  such 
a  chip  is  not  carried  on  by  the  rings  of  wave.  It  only  bobs  up  and  down  and 
finally  rests  on  its  original  spot.  The  adjacent  particles  of  water  move  in  the  same 
manner.  When  the  ring  of  wave  reaches  them  they  are  set  bobbing ;  when  it  has 
passed  over  them  they  are  still  in  their  old  place,  and  remain  there  at  rest,  while 
the  ring  of  wave  continues  to  advance  towards  fresh  spots  on  the  surface  of  the 
water,  and  sets  new  particles  of  water  in  motion.  Hence  the  waves  which  pass 
over  the  surface  of  the  water  are  constantly  built  up  of  fresh  particles  of  water. 
What  really  advances  as  a  wave  is  only  the  tremor,  the  altered  form  of  the  surface, 
while  the  individual  particles  of  water  themselves  merely  move  up  and  down  ^ 
transiently,  and  never  depart  far  from  their  original  position. 

The  same  relation  is  seen  still  more  clearly  in  the  waves  of  a  rope  or  chain. 
Take  a  flexible  string  of  several  feet  in  length,  or  a  thin  metal  chain,  hold  it  at  one 
end  and  let  the  other  hang  down,  stretched  by  its  own  weight  alone.  Now,  move 
the  hand  by  which  you  hold  it  quickly  to  one  side  and  back  again.  The  excursion 
which  we  have  caused  in  the  upper  end  of  the  string  by  moving  the  hand,  will  run 
down  it  as  a  kind  of  wave,  so  that  constantly  lower  parts  of  the  string  will  make  a 
sidewards  excursion  while  the  upper  return  again  into  the  straight  position  of  rest. 
But  it  is  evident  that  while  the  wave  runs  down,  each  individual  particle  of  the 
string  can  have  only  moved  horizontally  backwards  and  forwards,  and  can  have 
taken  no  share  at  all  in  the  advance  of  the  wave. 

The  experiment  succeeds  still  better  with  a  long  elastic  line,  such  as  a  thick 
piece  of  india-rubber  tubing,  or  a  brass-wire  spiral  spring,  from  eight  to  twelve  feet 
in  length,  fastened  at  one  end,  and  slightly  stretched  by  being  held  with  the  hand  ^ 
at  the  other.  The  hand  is  then  easily  able  to  excite  waves  wliich  will  run  very 
regularly  to  the  other  end  of  the  line,  be  there  reflected  and  return.  In  this  case 
it  is  also  evident  that  it  can  be  no  part  of  the  line  itself  which  runs  backwards  and 
forwards,  but  that  the  advancing  wave  is  composed  of  continually  fresh  particles 
of  the  line.  By  these  examples  the  reader  will  be  able  to  form  a  mental  image  of 
the  kind  of  motion  to  which  sound  belongs,  where  the  material  particles  of  the 
body  merely  make  periodical  oscillations,  while  the  tremor  itself  is  constantly 
propagated  forwards. 

Now  let  us  return  to  the  surface  of  the  water.  We  have  supposed  that  one  of 
its  points  has  been  struck  by  a  stone  and  set  in  motion.  This  motion  has  spread 
out  in  the  form  of  a  ring  of  wave  over  the  surface  of  the  water,  and  having  reached 
the  chip  of  wood  has  set  it  bobbing  iip  and  down.  Hence  by  means  of  the  wave, 
the  motion  which  the  stone  first  excited  in  one  point  of  the  surface  of  the  water 
has  been  communicated  to  the  chip  which  was  at  another  point  of  the  same  surface. 


10  FORCE,  PITCH,  AND  QUALITY.  part  i. 

The  process  which  goes  on  in  the  atmospheric  ocean  about  us,  is  of  a  precisely 
similar  nature.  For  the  stone  substitute  a  sounding  body,  which  shakes  the  air ; 
for  the  chip  of  wood  substitute  the  human  ear,  on  which  impinge  the  waves  of  air 
excited  by  the  shock,  setting  its  movable  parts  in  vibration.  The  waves  of  air 
proceeding  from  a  sounding  body,  transport  the  tremor  to  the  human  ear  exactly 
in  the  same  way  as  the  water  transports  the  tremor  produced  by  the  stone  to  the 
floating  chip. 

In  this  way  also  it  is  easy  to  see  how  a  body  which  itself  makes  periodical 
oscillations,  will  necessarily  set  the  particles  of  air  in  periodical  motion.  A  falling 
stone  gives  the  surface  of  the  water  a  single  shock.  Now  replace  the  stone  by  a 
regular  series  of  drops  falling  from  a  vessel  with  a  small  orifice.  Every  separate 
drop  will  excite  a  ring  of  wave,  each  ring  of  wave  will  advance  over  the  surface  of 
the  water  precisely  like  its  predecessor,  and  will  be  in  the  same  way  followed  by 

^  its  successors.  In  this  manner  a  regular  series  of  concentric  rings  will  be  formed 
and  propagated  over  the  surface  of  the  water.  The  number  of  drops  which  fall 
into  the  water  in  a  second  will  be  the  number  of  waves  which  reach  our  floating 
chip  in  a  second,  and  the  number  of  times  that  this  chip  will  therefore  bob  up  and 
down  in  a  second,  thus  executing  a  periodical  motion,  the  period  of  which  is  equal 
to  the  interval  of  time  between  the  falling  of  consecutive  drops.  In  the  same  way 
for  the  atuiosphere,  a  periodically  oscillating  sonorous  body  produces  a  similar 
periodical  motion,  first  in  the  mass  of  air,  and  then  in  the  drumskin  of  our  ear, 
and  the  period  of  these  vibrations  must  be  the  same  as  that  of  the  vibration  in  the 
sonorous  body. 

Having  thus  spoken  of  the  principal  division  of  sound  into  Noise  and  Musical 
Tones,  and  then  described  the  general  motion  of  the  air  for  these  tones,  we  pass 
on  to  the  peculiarities  which  distinguish  such  tones  one  from  the  othei*.  We  are 
acquainted  with  three  points  of  difference  in  musical  tones,  confining  oiu'  attention 

m  in  the  first  place  to  such  tones  as  are  isolatedly  produced  by  our  usual  musical 
instruments,  and  excluding  the  sinudtaneous  sounding  of  the  tones  of  different 
instruments.     Musical  tones  are  distinguished  : — 

1.  By  their  force, 

2.  By  their  2jifc/i, 

3.  By  their  qtialiti/. 

It  is  unnecessary  to  explain  what  we  mean  by  the  force  and  pitch  of  a  tone. 
By  the  (piality  of  a  tone  we  mean  that  peculiarity  which  distinguishes  the  musical 
tone  of  a  violin  from  that  of  a  flute  or  that  of  a  clarinet,  or  that  of  the  hiunan 
voice,  when  all  these  instruments  produce  the  same  note  at  the  same  pitch. 

We  have  now  to  explain  what  peculiarities  of  the  motion  of  sound  correspond 
to  these  three  principal  diftcrences  between  musical  tones. 

First,  We  easily  recognise  that  the  force  of  a  musical  tone  increases  and  dimi- 
nishes with  the  extent  or  so-called  amplitude  of  the  oscillations  of  the  particles  of 
■T  the  sounding  body.  When  we  strike  a  string,  its  vibrations  are  at  first  sufficiently 
large  for  us  to  see  them,  and  its  corresponding  tone  is  loudest.  The  visible 
vibrations  become  smaller  and  smaller,  and  at  the  same  time  the  loudness 
diminishes.  The  same  observation  can  be  made  on  strings  excited  by  a  violin 
bow,  and  on  the  reeds  of  reed-pipes,  and  on  many  other  sonorous  bodies.  The 
same  conclusion  results  from  the  diminution  of  the  loudness  of  a  tone  when  we 
increase  our  distance  from  the  sounding  body  in  the  open  air,  although  the  pitch 
and  quality  remain  unaltered ;  for  it  is  only  the  amplitude  of  the  oscillations  of 
the  particles  of  air  which  diminishes  as  their  distance  from  the  sounding  body 
increases.  Hence  loudness  must  depend  on  this  amplitude,  and  none  other  of  the 
properties  of  sound  do  so.* 

*  Mechanically  the  force  of  the  oscillations  no  measure  can  be  found  for  the  intensity  of 

for  tones  of  different  pitch   is   measured   by  the  sensation  of  sound,  that  is,  for  the  loudness 

their  vis  viva,  that  is,  by  the  square  of  the  of  sound  which  will  hold  all  pitches.     [See 

greatest  velocity  attained  by  the   oscillating  the  addition  to  a  footnote  on  p.  75f/,  referring 

particles.     But  the  ear  has  different  degrees  of  especially  to  this  passage.  —  Translator.] 
sensibility  for  tones  of  different  pitch,  so  that 


1 


PITCH  AND   THE  SIREN. 


The  second  essential  difference  between  difterent  musical  tones  consists  in 
their  J) if c/i.  Daily  experience  shows  iis  that  mnsical  tones  of  the  same  pitch  can 
be  prodnced  upon  most  diverse  instruments  by  means  of  most  diverse  mechanical 
contrivances,  and  with  most  diverse  degrees  of  loudness.  All  the  motions  of  tlie 
air  thus  excited  must  be  periodic,  because  they  would  not  otlierwise  excite  in  us 
the  sensation  of  a  nmsical  tone.  But  the  sort  of  motion  within  each  single 
period  may  be  any  whatever,  and  yet  if  the  length  of  the  periodic  time  of  two 
musical  tones  is  the  same,  they  have  the  same  pitch.  Hence :  Fitrk  (Ujwnih 
solely  on  the  length  of  time  in  which  each  single  vibration  is  executed,  or,  which 
comes  to  the  same  thing,  on  the  number  of  vibrations  completed  in  a  given  time. 
We  are  accustomed  to  take  a  second  as  the  unit  of  time,  and  shall  consequently 
mean  by  the  pitch  number  [or  frequoici/]  of  a  tone,  the  number  of  vibrations  which 
the  particles  of  a  sounding  body  perform  in  one  second  of  time.*  It  is  self-evident 
that  we  find  the  periodic  time  or  vibrational  period,  that  is  length  of  time  which  H 
is  occupied  in  performing  a  single  vibration  backwards  and  forwards,  by  dividing- 
one  second  of  time  by  the  pitch  number. 

Musical  tones  are  said  to  be  higher,  the  greater  their  pi  frit  numbers,  f/i<(t  is, 
the  shorter  their  vibrational  periods. 

The  exact  determination  of  the  pitch  niuuber  for  such  elastic  bodies  as  produce 
audible  tones,  presents  considerable  difficulty,  and  physicists  had  to  contrive  many 
comparatively  complicated  processes  in  order  to  solve  this  problem  for  each 
particular  case.  Mathematical  theory  and  numerous  experiments  had  to  render 
mutual  assistance. t  It  is  consequently  very  convenient  for  the  demonstration  of 
the  fundamental  facts  in  this  department  of  knowledge,  to  be  able  to  apply  a 
peculiar  instrument  for  producing  musical  tones — the  so-called  siren — which  is 
constructed  in  such  a  manner  as  to  determine  the  pitch  number  of  tlie  tone 
produced,  by  a  direct  observation.  The  principal  parts  of  the  simplest  form  of 
the  siren  are  shown  in  fig.  1,  after  Seebeck.  ^ 

A  is  a  thin  disc  of  cardboard  or  tinplate,  which  can  be  set  in  rapid  rotation 
about  its  axle  b  by  means  of  a  string  f  f,  which  passes  over  a  larger  wheel.  On 
the  margin  of  the  disc  there  is  punched  a  set  of  holes  at  equal  intervals  :  of  these 

there  are  twelve  in  the  figure  ;  one  or 
more  similar  series  of  holes  at  equal 
distances  are  introduced  on  concentric 
circles  (there  is  one  such  of  eight  holes 
in  the  figure),  c  is  a  pipe  which  is 
directed  over  one  of  the  holes.  Now, 
on  setting  the  disc  in  rotation  and  blow- 
ing through  the  pipe  c,  the  air  will  pass 
freely  whenever  one  of  the  holes  comes 
under  the  end  of  the  pipe,  but  will  be 
checked  whenever  an  unpierced  portion  ^ 
of  the  disc  comes  under  it.  Each  hole 
of  the  disc,  then,  that  passes  the  end  of  the  pipe  lets  a  single  puft"  of  air  escape. 
Supposing  the  disc  to  make  a  single  revolution  and  the  pipe  to  be  directed  to  the 


*  [The  pitch  number  was  called  the  '  vibra- 
tional number'  in  the  first  edition  of  this  trans- 
lation. The  pitch  n  umber  of  a  note  is  commonly 
called  the  pitch  of  the  note.  By  a  convenient 
abbreviation  we  often  write  a'  440,  meaning 
the  note  a'  having  the  i^itch  number  440 ;  or 
say  that  the  pitch  of  a'  is  440  vib.  that  is,  440 
double  vibrations  in  a  second.  The  second 
texxn.  frequency,  which  I  have  introduced  into 
the  text,  as  it  is  much  used  by  acousticians, 
properly  represents  Ihc  number  of  times  that 
any  periodically  recurrinq  event  happens  in 
one  second  of  time,  and,  applied  to  double 
vibrations,  it  means  the  same  as  pitch  number. 


The  pitch  of  a  musical  instrument  is  the  pitch 
of  the  note  by  wliich  it  is  tuned.  But  as  i^itch 
is  properly  a  sensation,  it  is  necessary  here 
to  distinguish  from  this  sensation  the  pitch 
number  or  frcq^icncy  of  vibration  by  which  it 
is  measured.  The  larger  the  pitch  number, 
the  higher  or  sharper  the  pitch  is  said  to  be. 
The  lower  the  pitch  nmnber  the  deeper  or 
flatter  the  pitch.  These  are  all  metaphorical 
expressions  which  must  be  taken  strictly  in 
this  sense. — Translator.'] 

t  [An  account  of  the  more  exact  modern 
methods  is  given  in  App.  XX.  sect.  B. — 
Ti-anslator.'] 


12 


PITCH  AND  THE  SIREN. 


outer  circle  of  holes,  we  have  twelve  puffs  corresponding  to  the  twelve  holes;  but 
if  the  pipe  is  directed  to  the  inner  circle  we  have  only  eight  puffs.  If  the  disc  is 
made  to  revolve  ten  times  in  one  second,  the  outer  circle  will  produce  120  puffs, 
in  one  second,  which  would  give  rise  to  a  v/eak  and  deep  musical  tone,  and  the 
inner  circle  eighty  puff's.  Generally,  if  we  know  the  number  of  revolutions  which 
the  disc  makes  in  a  second,  and  the  number  of  holes  in  the  series  to  which  the 
tube  is  directed,  the  product  of  these  two  numbers  evidently  gives  the  number  of 
puff's  in  a  second.  This  number  is  consequently  far  easier  to  determine  exactly 
than  in  any  other  musical  instrument,  and  sirens  are  accordingly  extremely  well 
adapted  for  studying  all  changes  .in  musical  tones  resulting  from  the  alterations 
and  ratios  of  the  pitch  numbers. 

The  form  of  siren  here  desci-ibed  gives  only  a  weak  tone.     I  have  placed  it  first 
because  its  action  can  be  most  readily  understood,-  and,  by  chana'inu'  the  disc,  it 


can  be  easily  applied  to  experiments  of  very  different  descriptions.  A  stronger  tone 
is  produced  in  the  siren  of  Cagniard  de  la  Tour,  shown  in  figures  2,  3,  and  4,  above. 
Here  s  s  is  the  rotating  disc,  of  which  the  upper  surface  is  shown  in  fig.  3,  and 
the  side  is  seen  in  figs.  2  and  4.  It  is  placed  over  a  windchest  A  A,  which  is 
connected  with  a  bellows  by  the  pipe  B  B.  The  cover  of  the  windchest  A  A, 
which  lies  immediately  under  the  rotating  disc  s  s,  is  pierced  with  precisely  the 
same  number  of  holes  as  the  disc,  and  the  direction  of  the  holes  pierced  in  the 
cover  of  the  chest  is  oblique  to  that  of  the  holes  in  the  disc,  as  shown  in  fig.  4, 
which  is  a  vertical  section  of  the  instrument  through  the  line  n  n  in  fig.  3.  This 
position  of  the  holes  enables  the  wind  escaping  from  A  A  to  set  the  disc  s  s  in 
rotation,  and  by  increasing  the  pressure  of  the  bellows,  as  much  as  50  or  60 
rotations  in  a  second  can  be  produced.  Since  all  the  holes  of  one  circle  are  blown 
through  at  the  same  time  in  this  siren,  a  much  more  powerful  tone  is  produced 
than  in  Seebeck's,  fig.   1   (p.   lie).      To  record   the  revolutions,  a  counter  z  z  is 


CHAP.  I.  PITCH  AND  INTERVAL.  13 

introducsd,  C3iiaeste:l  with  a  toothei  wheel  which  works  in  the  screw  t,  ;uid 
advances  one  tooth  for  each  revohitioii  of  the  disc  s  s.  By  the  handle  h  this 
counter  niav  be  moved  slightly  to  one  side,  so  that  the  wheelwork  and  screw  may 
be  connected  or  disconnected  at  pleasure.  If  they  are  connected  at  the  beginning 
of  one  second,  and  disconnected  at  the  beginning  of  another,  the  hand  of  the 
counter  shows  how  many  revolutions  of  the  disc  have  been  mide  in  the  corre- 
sponding number  of  seconds.* 

Dove+  introduced  into  this  siren  several  rows  of  holes  through  which  the  wind 
might  be  directed,  or  from  which  it  might  be  cut  oflf,  at  pleasure.  A  polyphonic 
siren  of  this  description  with  other  peculiar  arrangements  will  be  figured  and 
described  in  Chapter  VIII.,  fig.  56. 

It  is  clear  that  when  the  pierced  disc  of  one  of  these  sirens  is  made  to  revolve 
with  a  uniform  velocity,  and  the  air  escapes  through  the  holes  in  puffs,  the  motion 
of  the  air  thus  produced  must  be  x)eriod.k  in  the  sense  already  explained.  TheH 
holes  stand  at  equal  intervals  of  space,  and  hence  on  rotation  follow  each  other  at 
equal  intervals  of  time.  Through  every  hole  there  is  poured,  as  it  were,  a  drop  of 
air  into  the  external  atmospheric  ocean,  exciting  waves  in  it,  which  succeed  each 
other  at  uniform  intervals  of  time,  just  as  was  the  case  when  regulaidy  falling 
drops  impinged  upon  a  surface  of  water  (p.  lOa).  Within  each  separate  period, 
each  individual  pufF  will  have  considerable  variations  of  form  in  sirens  of  difterent 
construction,  depending  on  the  different  diameters  of  the  holes,  their  distance  from 
each  other,  and  the  shape  of  the  extremity  of  the  pipe  which  conveys  the  air ;  but 
in  every  case,  as  long  as  the  velocity  of  rotation  and  the  position  of  the  pipe  remain 
unaltered,  a  regulaidy  periodic  motion  of  the  air  must  result,  and  consequently  the 
sensation  of  a  musical  tone  must  be  excited  in  the  ear,  and  this  is  actually  the 
case. 

It  results  immediately  from  experiments  with  the  siren  that  two  series  of  the 
same  number  of  holes  revolving  with  the  same  velocity,  give  musical  tones  of  the  ^ 
same  pitch,  quite  independently  of  the  size  and  form  of  the  holes,  or  of  the  pipe. 
We  even  obtain  a  musical  tone  of  the  same  pitch  if  we  allow  a  metal  point  to 
strike  in  the  holes  as  they  revolve  instead  of  blowing.  Hence  it  follows  firstly  that 
the  pitch  of  a  tone  depends  only  on  the  Huinb<n-  of  puffs  or  swings,  and  not  on 
their  form,  force,  or  method  of  production.  Further  it  is  very  easily  seen  with 
this  instrument  that  on  increasing  the  velocity  of  rotation  and  consequently  the 
number  of  puffs  produced  in  a  second,  the  pitch  becomes  sharper  or  higher.  The 
same  result  ensues  if,  maintaining  a  uniform  velocity  of  rotation,  we  first  blow  into 
a  series  with  a  smaller  and  then  into  a  series  with  a  greater  niimber  of  holes. 
The  latter  gives  the  sharper  or  higher  pitch. 

With  the  same  instrument  we  also  very  easily  find  the  remarkable  relation 
which  the  pitch  numbers  of  two  musical  tones  must  possess  in  order  to  form  a 
consonant  interval.  Take  a  series  of  8  and  another  of  16  holes  in  a  disc,  and 
blow  into  both  sets  while  the  disc  is  kept  at  uniform  velocity  of  rotation.  TwoH 
tones  will  be  heard  which  stand  to  one  another  in  the  exact  relation  of  an  Octave. 
Increase  the  velocity  of  rotation ;  both  tones  will  become  sharper,  but  both  will 
continue  at  the  new  pitch  to  form  the  interval  of  an  Octave. J  Hence  we  conclude 
that  a  musical  tone  which  is  an  Octave  higher  than  another,  inaki^s  exaetli/  twice 
as  manij  vibrations  in  a  given  time  as  the  latter. 

*  See  Appendix  I.  names  of  all  the  intervals  usually  distinguished 

t  [Pronounce  Doh-reh,  in  two  syllables. —  are  also  given  in  App.  XX.  sect.  I).,  with  the 

Transtntor.]  corresponding  ratios  and  cents.     These  names 

\  [When   two   notes   have   different   pitch  were  in  the  first  place  derived  from  the  ordinal 

numbers,    there    is    said    to    be    an    interval  number  of  the  note  in  the  scales,  or  succes- 

between  them.      This  gives  rise  to  a  sensa-  sions  of  continually  sharper  notes.    The  Octave 

tion,  very  differently  appreciated  by  different  is  the  eighth  note  in  the  major  scale.    An  octave 

individuals,  but   in   all   cases  the  interval  is  is  a  set  of  notes  lying  within  an  Octave.     Ob- 

measured   by   the   ratio   of  the    pitch   mimbers,  serve  that  in  this  translation  all  names  of  in- 

and,  for  some  purposes,  more  conveniently  by  tervals   commence   with   a   capital   letter,  to 

other  numbers  called  cents,  derived  from  these  prevent  ambiguity,  as  almost  all  such  v/ords 

ratios,  as  explained  in  App.  XX.  sect.  C.     The  are  also  used  in  other  senses.— Translator.] 


U  PITCH  AND  INTERVAL.  part  i. 

The  disc  shown  m  fig.  1,  p.  11  r,  has  two  circles  of  8  and  12  holes  respectively. 
Each,  blown  successiveh',  gives  two  tones  which  form  with  each  other  a  perfect 
Fifth,  independently  of  the  velocity  of  rotation  of  the  disc.  Hence,  two  musical 
tones  stand  in  the  relation  of  a  so-called  Fifth  irJien  the  highe)-  tone  makes  three 
vibrations  in  the  same  time  as  the  lower  makes  two. 

If  we  obtain  a  musical  tone  by  blowing  into  a  circle  of  8  holes,  we  require  a 
circle  of  16  holes  for  its  Octave,  and  12  for  its  Fifth.  Hence  the  ratio  of  the 
pitch  numbers  of  the  Fifth  and  the  Octave  is  12  :  16  or  3  :  4.  But  the  interval 
between  the  Fifth  and  the  Octave  is  the  Fourth,  so  that  we  see  that  when  two 
musical  tones  form  a  Fourth,  the  higher  makes  four  vibrations  irhile  the  lower 
makes  three. 

The  polyphonic  siren  of  Dove  has  ixsually  four  circles  of  8,  10,  12  and  16  holes 
respectively.  The  series  of  16  holes  gives  the  Octave  of  the  series  of  8  holes,  and 
U  the  Fourth  of  the  series  of  1 2  holes.  The  series  of  1 2  holes  gives  the  Fifth  of  the 
series  of  8  holes,  and  the  minor  Third  of  the  series  of  10  holes.  While  the  series  of 
10  holes  gives  the  major  Third  of  the  series  of  8  holes.  The  four  series  con- 
sequently give  the  constituent  musical  tones  of  a  major  chord. 

By  these  and  similar  experiments  we  find  the  following  relations  of  the  pitch 
numbers  : — 

1  :    2     Octave 

2  :    3     Fifth 

3  :    4     Fourth 

4  :    5     major  Third 

5  :    6     minor  Third 

When  the  fundamental  tone  of  a  given  interval  is  taken  an  Octave  higher,  the 
interval  is  said  to  be  inverted.     Thus  a  Fourth  is  an  inverted  Fifth,  a  minor  Sixth 
^  an  inverted  major  Third,  and  a  major  Sixth  an  inverted  minor  Third.     The  corre- 
sponding ratios  of  the  pitch  numbers  are  consequently  obtained  by  doubling  the 
smaller  number  in  the  original  interval. 

From  2  :  3     the  Fifth,  we  thus  have        3  :  4     the  Fourth 
„      4:5     the  major  Third       ...       5:8     the  minor  Sixth 
„      5:6     the  minor  Third,     6  :  10=  3  :  5     the  major  Sixth. 

These  are  all  the  consonant  intervals  which  lie  within  the  compass  of  an 
Octave.  With  the  exception  of  the  minor  Sixth,  which  is  really  the  most  imperfect 
of  the  above  consonances,  the  ratios  of  their  vibrational  numbers  are  all  expressed 
by  means  of  the  whole  numbers,  1,  2,  3,  4,  5,  6. 

Comparatively  simple  and  easy  experiments  with  the  siren,  therefore,  corrobo- 
rate that  remarkable  law  mentioned  in  the  Introduction  (p.  Id),  according  to  which 
the  pitch  numbers  of  consonant  musical  tones  bear  to  each  other  ratios  expressible 
H  by  small  whole  numbers.  In  the  course  of  our  investigation  we  shall  employ  the 
same  instrument  to  verify  more  com})letely  the  strictness  and  exactness  of  this 
law. 

Long  before  anything  was  known  of  pitch  numbers,  or  the  means  of  counting 
them,  Pythagoras  had  discovered  that  if  a  string  be  divided  into  two  parts  by  a 
bridge,  in  such  a  way  as  to  give  two  consonant  musical  tones  when  struck,  the 
lengths  of  these  parts  must  be  in  the  ratio  of  these  whole  numbers.  If  the  bridge 
is  so  placed  that  f  of  the  string  lie  to  the  right,  and  ~  on  the  left,  so  that  the  two 
lengths  are  in  the  ratio  of  2  :  1,  they  produce  the  interval  of  an  Octave,  the  greater 
length  giving  the  deeper  tone.  Placing  the  bridge  so  that  f  of  the  string  lie  on 
the  right  and  f  on  the  left,  the  ratio  of  the  two  lengths  is  3  :  2,  and  the  interval 
is  a  Fifth. 

These  measurements  had  been  executed  with  great  precision  by  the  Creek 
musicians,  and  had  given  rise  to  a  system  of  tones,  contrived  with  considerable 
art.     For   these   measurements  they  used   a  peculiar   instnunent,  the   monochord. 


CHAP.  I.  PITCH  NUMBERS  IN  JUST  MAJOR  SCALE.  15 

consisting  of  a  sonnding  board  and   box  on  which    a   single  string  was  stretched 
with  a  scale  below,  so  as  to  set  the  bridge  correctly.* 

It  was  not  till  ninch  later  that,  through  the  investigations  of  CJalileo  (1638), 
Newton,  Euler  (1729),  and  Daniel  Bernouilli  (1771),  the  law  governing  the 
motions  of  strings  became  known,  and  it  was  thus  found  that  the  simple  ratios  of 
the  lengths  of  the  strings  existed  also  for  the  pitch  numbers  of  the  tones  they  pro- 
duced, and  that  they  consequently  belonged  to  the  musical  intervals  of  the  tones 
of  all  instruments,  and  were  not  confined  to  the  lengths  of  strings  through  which 
the  law  had  been  first  discovered. 

This  relation  of  whole  numbers  to  musical  consonances  was  from  all  time 
looked  upon  as  a  wonderful  mystery  of  deep  significance.  Tlie  Pythagoreans 
themselves  made  use  of  it  in  their  speculations  on  the  harmony  of  the  spheres. 
From  that  time  it  remained  partly  the  goal  and  partly  the  starting  point  of  the 
strangest  and  most  venturesome,  fantastic  or  philosophic  combinations,  till  in  ^ 
modern  times  the  majority  of  investigators  adopted  the  notion  accepted  by  Eider 
himself,  that  the  human  mind  had  a  peculiar  pleasure  in  simple  ratios,  because  it 
could  better  understand  them  and  comprehend  their  bearings.  But  it  remained 
uninvestigated  how  the  mind  of  a  listener  not  versed  in  physics,  who  perhaps  was 
not  even  aware  that  musical  tones  depended  on  periodical  vibrations,  contrived  to 
recognise  and  compare  these  ratios  of  the  pitch  numbers.  To  show  what  pro- 
cesses taking  place  in  the  ear,  render  sensible  the  diflference  between  consonance 
and  dissonance,  will  be  one  of  the  principal  problems  in  the  second  part  of  this 
work . 

Calculatiox  of  the  Pitch  Numbers  for  all  the  Tones  of  the 
Musical  Scale. 

By  means  of  the  ratios  of  the  pitch  numbers  already  assigned  for  the  consonant 
intervals,  it  is  easy,  by  pursuing  these  intervals  throughout,  to  calculate  the  ratios  ^ 
for  the  whole  extent  of  the  musical  scale. 

The  major  triad  or  chord  of  three  tones,  consists  of  a  major  Third  and  a  Fifth. 
Hence  its  ratios  are  : 

C  :  E  :  G 

1  :  A  :^ 

or    4:5:  6 

If  we  associate  with  this  triad  that  of  its  dominant  G  :  B  :  D,  and  that  of  its 
sub-dominant  F  :  A  :  C,  each  of  which  has  one  tone  in  common  with  the  triad  of 
the  tonic  C  :  E  :  G,  we  obtain  the  complete  series  of  tones  for  the  major  scale  of 
C,  with  the  following  ratio  of  the  pitch  numbers  : 

C  :  D  :  E  \  F  :  G  :  A  \  B  :  v 

[or     24  :  27  :  30  :  32  :  36  :  40  :  45  :  48]  ^I 

In  order  to  extend  the  calculation  to  other  octaves,  we  shall  adopt  the  following 
notation  of  musical  tones,  marking  the  higher  octaves  by  accents,  as  is  usual  in 
(ilermany,t  as  follows : 

1.   The  unaccented  or  small  octave  (the  4-foot  octave  on  the  organj)  :  — 


c            d            e  f            (J            a            h 

*  [As  the  monochord  is  very  liable  to  error,  below  the  letters,  which  are  typographically 

these  results  were  happy  generalisations  from  inconvenient.     Hence  the  German  notation  is 

necessarily    imperfect    experiments. — Tmns-  retained. — Translator.] 

lator.]  +  [The   note    C  in   the   small   octave  was 

t  [English   works   use   strokes   above  and  once  emitted  by  an  organ  pipe  4  feet  in  length : 


16  PITCH  NUMBERS  IN  JUST  MAJOR  SCALE.  pari 

2.    IVie  once-accented  octave  (2-foot)  : — 


^._ ^ ^ 

c'  d'  e'  f 

3.   The  twice-accented  octave  (1-foot)  : — 


*^        c"  d"  e"  f"  <j"  a"  h" 

And   so  on   for  higher  octaves.      Below   the   small   octave   lies   the  great    octave, 
written  with  unaccented  capital  letters  ;  its  C  requires  an  organ  pipe  of  eight  feet 
H  in  length,  and  hence  it  is  called  the  8-foot  octave. 

4.   Great  or  8-foot  octave  : — 


"O" 


-&- 


-^ &- 


C         D  E         F  G  A  B 

Below  this  follows  the  KS-foot  or  contra-octave ;  the  lowest  on  the  pianoforte 
and  most  organs,  the  tones  of  which  may  be  represented  by  C ^  D ^  E ^  F ^  G ^  A^  B ., 
with  an  inverted  accent.  On  great  organs  there  is  a  still  deeper,  32-foot  octave,  the 
tones  of  which  may  be  written  C ^^  D ^^  E^^  F^^  G^^  A  ^^  B^,  with  two  inverted  accents, 
but  they  scarcely  retain  the  character  of  musical  tones.     (See  Chap.  JX.) 

Since  the  pitch  numbers  of  any  octave  are  always  twice  as  great  as  those  for 
11  the  next  deeper,  we  find  the  pitch  numbers  of  the  higher  tones  by  multiplying 
those  of  the  small  or  unaccented  octave  as  many  times  by  2  as  its  symbol  has 
upper  accents.  And  on  the  contrary  the  pitch  numbei's  for  the  deeper  octaves  are 
found  by  dividing  those  of  the  great  octave,  as  often  as  its  symbol  has  lower 
accents. 

Thus  (•"  =  2x2xc=2x2x2C 
C.  =  i  X  i  X  <^'  =  i  X  *  X  i  c. 

For  the  pitch  of  the  musical  scale  German  physicists  have  generally  adopted 
that  proposed  by  Scheibler,  and  adopted  siibseqiiently  by  the  German  Association 
of  Natural  Philosophers  {die  deutsche  Naturforscherversammhing)  in  1834.  Tliis 
makes  the  once-accented  a  execute  440  vibrations  in  a  second.*     Hence  results  the 

thus  Bedos  {L'Art  du  Fadcur  d'Orgues,  1766)  backward  and  forward  niotiois  it  would  be 
51  made  it  4  old  French  feet,  which  gave  a  indifferent  by  which  method  we  counted,  but 
note  a  full  Semitone  flatter  than  a  pipe  of  for  non-symmetrical  musical  vibrations  which 
4  English  feet.  But  in  modern  organs  not  even  are  of  constant  occurrence,  the  French  method 
so  much  as  4  English  feet  are  used.  Organ  of  counting  is  very  inconvenient.  The  number 
builders,  however,  in  all  countries  retain  the  440  gives  fewer  fractions  for  the  first  [just] 
nanres  of  the  octaves  as  here  given,  which  major  scale  of  C,  than  «'  =  435.  The  difference 
must  be  considered  merely  to  determine  the  of  pitcli  is  less  than  a  comma.  [The  practical 
place  on  the  staff,  as  noted  in  the  text,  inde-  settlement  of  pitch  has  no  relation  to  such 
pendently  of  the  precise  X3itch. —  Travslatnr.'\  arithmetical  considerations  as  are  here  sug- 
*  The  Paris  Academy  has  lately  fixed  the  gested,  but  depends  on  the  compass  of  the 
pitch  number  of  the  same  note  at  435.  This  human  voice  and  the  music  written  for  it  at 
is  called  870  by  the  Academy,  because  French  different  times.  An  Abstract  of  my  History 
physicists  have  adoj)ted  the  inconvenient  of  Musical  Fitch  is  given  in  Appendix  XX. 
habit  of  counting  the  forward  motion  of  a  sect.  H.  Scheibler's  proposal,  named  in  the 
swinging  body  as  one  vibration,  and  the  back-  text,  was  chosen,  as  he  tells  us  {Der  Tunmcsscr, 
ward  as  another,  so  that  the  whole  vibra-  1884,  p.  53),  as  being  the  mean  between  the 
tion  is  counted  as  two.  This  method  of  limits  of  pitch  within  which  Viennese  piano- 
counting  has  been  taken  from  the  seconds  fortes  at  that  time  rose  and  fell  by  heat  and 
pendulum,  which  ticks  once  in  going  forward  cold,  which  he  reckons  at  J  vibration  either 
and  once  again  on  returning.   For  symmetrical  way.  That  this  proposal  had  no  reference  to  the 


I 


CHAP.  I.  PITCH  NUMBERS  IN  JUST  :\IAJOR  SCALE.  17 

following  table  for  the  scale  of  C  major,  which  will  serve  to  determine  the  ])itch 
of  all  tones  that  are  defined  by  their  pitch  numbers  in  the  following  work. 


Contra  Octave 

Notes. 

C  toB, 

16  foot 

C 

33 

D 

37-125 

E 

41-25 

F 

44 

G 

49-5 

A 

55 

B 

61-875 

Great  Octave 
Cto  JS 

8  foot 


60 

74-25 

82-5 

88 

99 
110 
123-75 


Unaccented 
Octave 
c  to  6 
4  foot 


132 

148-5 

165 

170 

198 

220 

247-5 


Once- 
accented 
Octave 


264 
297 
330 
352 
396 
440 
495 


Twice- 
accented 
Octave 
c"  to  h" 
Ifoot 


528 
594 
660 
704 
792 


Thrice- 
accented 

Octave 
c"  to  b- 

ifoot 


1056 

1188 
1320 
1408 
1584 
1760 
1980 


Fovir-times 

accented 

Octave 

c'"  to  b'"' 

J  foot 


2112 
2376 
2640 
2816 
3168 
3520 
3960* 


The  lowest  tone  on  orchestral  instruments  is  the  E^  of  the  double  bass,  making 
Modern  pianofortes  and  organs  usually  go  down  to  C,  ^ 


expression  of  the  Just  major  scale  in  whole 
numbers,  is  shown  by  the  fact  that  he 
proposed  it  for  an  equally  tempered  scale, 
for  wbicli  he  calculated  the  pitch  numbers 
to  four  places  of  decimals,  and  for  which,  of 
course,  none  but  the  octaves  of  a'  are  ex- 
pressible by  whole  numbers. —  Translator.'] 

*  [As  itis  important  that  students  should 
be  able  to  hear  the  exact  intervals  and  pitches 
spoken  of  throughout  this  book,  and  as  it  is 
quite  impossible  to  do  so  on  any  ordinary  in- 
strument, I  have  contrived  a  specially-tuned 
harmonium,  called  an  Harmonical,  fully  de- 
scribed in  App.  XX.  sect.  F.  No.  1,  which 
Messrs.  Moore  &  Moore,  104  Bishopsgate  Street, 
will,  in  the  interests  of  science,  supply  to  order, 
for  the  moderate  sum  of  165i-.  The  follow- 
ing are  the  pitch  numbers  of  the  first  four 
octaves,  the  tuning  of  the  fifth  octave  will  be 


explained  in  App.  XX.  sect.  F.  The  names  of 
the  notes  are  in  tlie  notation  of  the  latter  part 
of  Chap.  XIV.  below.  Read  the  sign  i*,  as 
'/)  one,'  E^\)  as  'one  E  flat,'  and  'B\ji  as 
'  seven  B  flat '.  In  playing  observe  that  Z*,  is 
on  the  ordinary  D\f  or  G'|  digital,  and  that 
'/)'[j  is  on  the  ordinary  G\)  or  i*^  digital,  and 
that  the  only  keys  in  which  chords  can  be 
played  are  U  major  and  0  minor,  with  the 
minor  chovA  D^F A  ^  and  the  natural  chord  of 
the  Ninth  CE\G''B\jB.  The  mode  of  measuring 
intervals  by  ratios  and  cents  is  fully  explained 
hereafter,  and  the  results  are  added  for  con- 
venience of  reference.  The  pitches  of  c"  528, 
a'  440,  ft'i[j  422-4  and  'h'\y  462,  were  taken  from 
forks  very  carefully  tuned  by  myself  to  these 
numbers  of  vibrations,  by  means  of  my  unique 
series  of  forks  described  in  App.  XX.,  at  the  H 
end  of  sect.  B. 


Scale 

OF    THE    HaRMOMICAL. 

Pitch  Numbers, 

Ratios. 

Cents. 

Notes. 

Sfoot 

4  foot 

2  foot 

Ifoot 

Note  to  Note 

C  to  Note 

Note  to  Note 

C  to  Note 

6' 

66 

132 

264 

528 

9:  10 

1  :  1 

182 

0 

^1 

731 

1461 

2931 

5861 

80  :  81 

9  :  10 

22 

182 

D 

m 

148* 

297 

594 

15  :16 

8:9 

112 

204 

E'\> 

m 

1584 

316| 

633f 

24  :  25 

5  :  6 

70 

316 

E, 

82  .i 

165 

3.30 

660 

15  :  16 

4  :  5 

112 

386 

F 

88 

176 

352 

704 

8  :9 

3:4 

204 

498 

G 

99 

198 

396 

792 

15:  16 

2  :  3 

112 

702 

A'\y 

105j 

211i 

422| 

8444 

24  :  25 

5:  8 

70 

814 

^1 

110 

220 

440 

880 

20  :  21 

3:5 

85 

884 

-B'r, 

11.5^ 

231 

462 

924 

35  :  36 

4  :7 

49 

969 

B^b 

1184 

237f 

475i 

950| 

24  :  25 

5:9 

70 

1018 

B\ 

123= 

247* 

495 

990 

8:  15 

— 

1088 

15  :  16 

112 

c 

132       1     264 

528 

1056 

1  :  2 

— 

1200 

Translator.'] 


t  [The  following  account  of  the  actual  tones 
3d  is  adapted  from  my  History  of  Musical 


Fitch.     G  ,  commencement  of  the  32-foot  oc- 
tave, the 'lowest  tone  of  verj'  large  organs,  two 

C 


18 


COMPASS  OF  INSTRUMENTS. 


with  33  viljivations,  and  the  latest  grand  pianos  even  down  to  A^^  with  27^  vibra- 
tions. On  larger  organs,  as  already  mentioned,  there  is  also  a  deeper  Octave  reach- 
ing to  C„  with  16i  vibrations.  But  the  musical  character  of  all  these  tones  below  F^ 
is  imperfect,  because  we  are  here  near  to  the  limit  of  the  power  of  the  ear  to  combine 
vibrations  into  musical  tones.  These  lower  tones  cannot  therefore  be  vised  musically 
except  in  connection  with  their  higher  octaves  to  which  they  impart  a  character 
of  o-reater  depth  without  rendering  the  conception  of  the  pitch  indeterminate. 

Upwards,  pianofortes  generally  reach  a""  with  b520,  or  even  c"  with  4224  vibra- 
tions. The  highest  tone  in  the  orchestra  is  probably  the  five-times  accented  J"  of  the 
piccolo  flute  with  4752  vibrations.  Appunn  and  W.  Preyer  by  means  of  small 
tunino--forks  excited  by  a  violin  bow  have  even  reached  the  eight  times  accented  «="" 
with  40,960  vibrations  in  a  second.  These  high  tones  were  very  painfully  unplea- 
sant, and  the  pitch  of  those  which  exceed  the  boundaries  of  the  musical  scale  was 
^  very  imperfectly  discriminated  by  musical  observers."*     More  on  this  in  Chap.  IX. 

The  musical  tones  which  can  be  used  with  advantage,  and  have  clearly  dis- 
tinguishable pitch,  have  therefore  between  40  and  4000  vibrations  in  a  second, 
extendino-  over  7  octaves.  Those  which  are  audible  at  all  have  from  20  to  40,000 
vibrations,  extending  over  about  11  octaves.  This  shows  what  a  great  variety  of 
different  pitch  numbers  can  be  perceived  and  distinguished  by  the  ear.  In  this 
respect  the  ear  is  far  superior  to  the  eye,  which  likewise  distinguishes  light  of  dif- 
ferent periods  of  vibration  by  the  sensation  of  different  colours,  for  the  compass  of 
the  vibrations  of  light  distinguishable  by  the  eye  but  slightly  exceeds  an  Octave.t 

Fo7re  and  2}itch  were  the  two  first  differences  which  we  found  between  musical 
tones ;  the  third  was  quality  of  tone,  which  we  have  now  to   investigate.     When 

of  Tone,'  [ilbcr  die  Grcnzen  dcr  Tonv:ahrnc]i- 
muncj,  1876,  p.  20),  are  in  the  South  Kensing- 
ton Museum,  Scientific  Collection.  I  have 
several  times  tried  them.  I  did  not  myself 
find  the  tones  painful  or  cutting,  probably 
because  there  was  no  beating  of  inharmonic 
upper  ^Dartials.  It  is  best  to  sound  them  with 
two  violin  bows,  one  giving  the  octave  of  the 
other.  The  tones  can  be  easily  heard  at  a 
distance  of  more  than  100  feet  in  the  gallery 
of  the  Museum. — Translator.'] 

t  [Assuming  the  undulatory  theory,  which 
attributes  the  sensation  of  light  to  the  vibra- 
tions of  a  supposed  luminous  '  ether,'  resem- 
bling air  but  more  delicate  and  mobile,  then 
the  phenomena  of  '  interference '  enables  us 
to  calculate  the  lengths  of  waves  of  light  in 
empty  space,  &c. ,  hence  the  numbers  of  vibra- 
tions"in  a  second,  and  consequently  the  ratios 
of  these  numbers,  which  will  then  clearly 
resemble  the  ratios  of  the  pitch  nimibers  that 
measure  musical  intervals.  Assuming,  then, 
that  the  yellow  of  the  spectriun  answers  to  the 
tenor  c  in  music,  and  Fraunhofer's  '  line  A  ' 
corresponds  to  the  G  below  it,  Prof.  Helm- 
holtz,  in  his  Physiological  Optics,  {Hand- 
buch  der  physiologischen  Optik,  1867,  p.  237), 
gives  the  following  analogies  between  the  notes 
of  the  piano  and  the  colours  of  the  spectrmn  :— 


Octaves  below  the  lowest  tone  of  the  Violon- 
cello. A,„  the  lowest  tone  of  the  largest 
pianos.  C\,  commencement  of  the  16-foot 
octave,  the  lowest  note  assigned  to  the  Double 

U  Bass  in  Beethoven's  Pastoral  Symphony.  JS,, 
the  lowest  tone  of  the  German  four-stringed 
Double  Bass,  the  lowest  tone  mentioned  in 
the  text.  F„  the  lowest  tone  of  the  English 
four-stringed  Double  Bass.  G,,  the  lowest  tone 
of  the  Italian  three- stringed  Double  Bass.  A„ 
the  lowest  tone  of  the  English  three-stringed 
Double  Bass.  C,  conmiencement  of  the  8-foot 
octave,  the  lowest  tone  of  the  Violoncello, 
written  on  the  second  leger  line  below  the  bass 
stafi.  G,  the  tone  of  the  third  open  string  of 
the  Violoncello.  c,  commencement  of  the 
4-foot  octave  '  tenor  C,'  the  lowest  tone  of  the 
Viola,  written  on  the  second  space  of  the  bass 
staff,  d,  the  tone  of  the  second  open  string  of 
the  Violoncello.  /,  the  tone  signified  by  the 
bass  or  i^-clef.  ;/,  the  lowest  tone  of  the 
Violin,  a,  the  tone  of  the  highest  open  string 
of  the  Violoncello,     c',  conmiencement  of  the 

51  2-foot  octave,  '  middle  6','  written  on  the  leger 
line  between  the  bass  and  treble  staves,  the  tone 
signified  by  the  tenor  or  C-clef .  d',  the  tone  of  the 
third  open  string  of  the  Violin,  g',  the  tone 
signified  by  the  treble  or  G-clei.  a',  the  tone  of 
the  second  open  string  of  the  Violin,  the  'tuning 
note '  for  orchestras,  t",  commencement  of  the 
1-foot  octave,  the  usual '  tuning  note '  for  pianos. 
e",  the  tone  of  the  first  or  highest  open  string  of 
the  Violin,  c",  commencement  of  the  ^-foot 
octave,  g'",  the  usual  highest  tone  of  the 
Flute.  Civ,  commencement  of  the  ^-foot  octave. 
€'",  the  highest  tone  on  the  Violin,  being  the 
double  Octave  harmonic  of  the  tone  of  the 
highest  open  string,  a}"",  the  usual  highest 
tone  of  large  pianos.  tZ^',  the  highest  tone  of 
the  piccolo  flute.  c^"i,  the  highest  tone  reached 
by  Appunn's  forks,  see  next  note. —  Translator.} 
*  [Copies  of  these  forks,  described  in  Prof. 
Preyer's  essay '  On  the  Limits  of  the  Perception 


F 1  end  of  the  Red. 
G,Iied. 

f  i,  Violet. 
■(/,  Ultra-violet. 

G  i  Red. 
^,*Red. 

^t'         " 

((,„         .. 

A  5,  Orange-red. 
^,X>range. 

b,  end   of  the  solar 

spectrum. 

c.  Yellow. 

The  scale  there- 

c it.  Green. 

fZ,  Greenish-blue. 

fore    extends     to 

about    a    Fourth 

d  |,  Cyanogen-blue. 
e,   Indigo-blue. 

beyond    the    oc- 

tave. —  2'ransla- 

/,  Violet. 

tor.] 

CHAP.  I.  QUALITY  OF  TONE  AND  FORM  OF  VIBRATION.  19 

we  hear  notes  of  the  same  force  and  same  pitch  sonnded  snccessively  on  a  piano- 
forte, a  vioUn,  clarinet,  oboe,  or  trumpet,  or  by  the  liuman  voice,  the  character  of 
the  musical  tone  of  each  of  these  instruments,  notwithstanding  the  identity  of  force 
and  pitch,  is  so  different  that  by  means  of  it  we  recognise  witli  the  greatest  ease 
which  of  these  instruments  was  used.  Varieties  of  quality  of  tone  appear  to  be 
infinitely  numerous.  Not  only  do  we  know  a  long  series  of  musical  instruments 
which  could  each  produce  a  note  of  the  same  pitch ;  not  only  do  diflerent  individual 
instruments  of  the  same  species,  and  the  voices  of  different  individual  singers  show 
certain  more  delicate  shades  of  quality  of  tone,  which  our  ear  is  able  to  distinguish ; 
but  notes  of  the  same  pitch  can  sometimes  be  sounded  on  the  same  instrument  with 
several  qualitative  varieties.  In  this  respect  the  '  bowed '  instruments  (i.e.  those 
of  the  violin  kind)  are  distinguished  above  all  other.  But  the  human  voice  is  still 
richer,  and  human  speech  employs  these  very  qualitative  varieties  of  tone,  in  order 
to  distinguish  different  letters.  The  different  vowels,  namely,  belong  to  the  class  H 
of  sustained  tones  which  can  be  used  in  music,  while  the  character  of  consonants 
mainly  depends  upon  brief  and  transient  noises. 

On  inquiring  to  what  external  physical  difference  in  the  waves  of  sound  the 
different  qualities  of  tone  correspond,  we  must  remember  that  the  amplitude  of 
the  vibration  determines  the  force  or  loudness,  and  the  period  of  vibration  the 
pitch.  Quality  of  tone  can  therefore  depend  upon  neither  of  these.  The  only 
possible  hypothesis,  therefore,  is  that  the  quality  of  tone  should  depend  upon  the 
manner  in  which  the  motion  is  performed  within  the  period  of  each  single  vibra- 
tion. For  the  generation  of  a  musical  tone  we  have  only  required  that  the  motion 
should  be  periodic,  that  is,  that  in  any  one  single  period  of  vibration  exactly  the 
same  state  should  occur,  in  the  same  order  of  occurrence  as  it  presents  itself  in  any 
other  single  period.  As  to  the  kind  of  motion  that  should  take  place  within  any 
single  period,  no  hypothesis  was  made.  In  this  respect  then  an  endless  variety  of 
motions  might  be  possibly  for  the  production  of  sound.  ^ 

Observe  instances,  taking  first  such  periodic  motions  as  are  performed  so  slowly 
that  -we  can  follow  them  with  the  eye.  Take  a  pendulum,  which  we  can  at  any 
time  construct  by  attaching  a  weight  to  a  thread  and  setting  it  in  motion.  The 
pendulum  swings  from  right  to  left  with  a  imiform  motion,  uninterrupted  by  jerks. 
Near  to  either  end  of  its  path  it  moves  slowly,  and  in  the  middle  fast.  Among 
sonorous  bodies,  which  move  in  the  same  way,  only  very  much  faster,  we  may  . 
mention  tuning-forks.  When  a  tuning-fork  is  struck  or  is  excited  by  a  violin  bow, 
and  its  motion  is  allowed  to  die  away  slowly,  its  two  prongs  oscillate  backwards 
and  forwards  in  the  same  way  and  after  the  same  law  as  a  pendulum,  only  they 
make  many  hundred  swings  for  each  single  swing  of  the  pendulum. 

As  another  example  of  a  periodic  motion,  take  a  hammer  moved  by  a  water- 
wheel.  It  is  slowly  raised  by  the  millwork,  then  released,  and  falls  down  suddenly, 
is  then  again  slowly  raised,  and  so  on.  Here  again  we  have  a  periodical  backwards 
and  forwards  motion ;  but  it  is  manifest  that  this  kind  of  motion  is  totally  diflf'erent  ^ 
from  that  of  the  pendulum.  Among  motions  wdiich  produce  musical  sounds,  that  of 
a  violin  string,  excited  by  a  bow,  would  most  nearly  correspond  with  the  hammer's, 
as  will  be  seen  from  the  detailed  description  in  Chap.  V.  The  string  clings  for  a 
time  to  the  bow,  and  is  carried  along  by  it,  then  suddenly  releases  itself,  like  the 
hammer  in  the  mill,  and,  like  the  latter,  retreats  somewhat  with  much  greater 
velocity  than  it  advanced,  and  is  again  caught  by  the  bow  and  carried  forward. 

Again,  imagine  a  ball  thrown  up  vertically,  and  caught  on  its  descent  with  a 
blow  which  sends  it  up  again  to  the  same  height,  and  suppose  this  operation  to  be 
performed  at  equal  intervals  of  time.  Such  a  ball  would  occupy  the  same  time  in 
rising  as  in  falling,  but  at  the  lowest  point  its  motion  would  be  suddenly  interrupted, 
whereas  at  the  top  it  wovdd  pass  through  gradually  diminishing  speed  of  ascent 
into  a  gradually  increasing  speed  of  descent.  This  then  would  be  a  third  kind  of 
alternating  periodic  motion,  and  would  take  place  in  a  manner  essentially  different 
from  the  other  two. 

c  2 


20 


FORM  OF  VIBRATION. 


To  render  the  law  of  such  motions  more  comprehensible  to  the  eye  than  is 
le  by  lengthy  verbal  descriptions,  mathematicians  and  physicists  are  in  the 
habit  of  applying  a  graphical  method,  which  must  be  frequently  employed  in  this 
work,  and  should  therefore  be  well  understood. 

To  render  this  method  intelligible  suppose  a  drawing  point  b,  fig.  5,  to  be 
fastened  to  the  prong  A  of  a  tuning-fork  in  such  a  manner  as  to  mark  a  surface 
of  pauer  B  B.  Let  the  tuning-fork  be  moved  with  a  uniform  velocity  in  the  direc- 
tion of  the  upper  arrow,  or  else  the  paper  be  drawn  under  it  in  the  opposite 
direction,  as  shown  by  the  lower  arrow.  When  the  fork  is  not  sounding,  the  point 
will  describe  the  dotted  straight  line  d  c.  But  if  the  prongs  have  been  first  set  in 
vibration,  the  point  will  describe  the  undulating  line  d  c,  for  as  the  prong  vibrates, 
the  attached  point  b  will  constantly  move  backwards  and  forwards,  and  hence  be 


^ 


sometimes  on  the  right  and  sometimes  on  the  left  of  the  dotted  straight  line  d  c,  as 
is  shown  by  the  wavy  line  in  the  figure.  This  wavy  line  once  drawn,  remains  as  a 
permanent  image  of  the  kind  of  motion  performed  by  the  end  of  the  fork  during 
^  its  musical  vibrations.  As  the  point  b  is  moved  in  the  direction  of  the  straight 
line  d  c  with  a  constant  velocity,  equal  sections  of  the  straight  line  d  c  will  corre- 
spond to  equal  sections  of  the  time  during  which  the  motion  lasts,  and  the  distance 
of  the  wavy  line  on  either  side  of  the  straight  line  will  show  how  far  the  point  b 
has  moved  from  its  mean  position  to  one  side  or  the  other  during  those  sections  of 
time. 

In  actually  performing  such  an  experiment  as  this,  it  is  best  to  wrap  the  paper 
over  a  cylinder  which  is  made  to  rotate  uniformly  by  clockwork.  The  paper  is 
wetted,  and  then  passed  over  a  turpentine  flame  which  coats  it  with  lampblack, 
on  which  a  fine  and  somewhat  smooth  steel  point  will  easily  trace  delicate  lines. 


Fig.  6  is  the  copy  of  a  drawing  actually  made  in  this  way  on  the  rotating  cylinder 
of  Messrs.  Scott  and  Koenig's  Phonmttograph. 

Fig.  7  shows  a  portion  of  this  curve  on  a  larger  scale.  It  is  easy  to  see  the 
meaning  of  such  a  curve.  The  drawing  point  has  passed  with  a  uniform  velocity 
in  the  direction  e  h.  Suppose  that  it  has  described  the  section  e  g  in  -^^  of  a 
second.  Divide  e  g  into  12  equal  parts,  as  in  the  figure,  then  the  point  has  been 
y^o^  of  a  second  in  describing  the  length  of  any  such  section  horizontally,  and 
the  curve  shows  us  on  what  side  and  at  what  distance  from  the  position  of 
rest  the  vibrating  point  will  be  at  the  end  of  ■^~-^,  yf^,  and  so  on,  of  a  second, 
or,  generally,  at  any  given  short  interval  of  time  since  it  left  the  point  e. 
We  see,  in  the  figure,  that  after  yi^  of  a  second  it  had  reached  the  height  1, 
and  that  it  rose  gradually  till  the  end  of  yf  ^  of  a  second ;  then,  however,  it  began 
to  descend  gradually  till,  at  the  end  of  Tfo  =  oV  seconds,  it  had  reached  its  mean 


FORM  OF  VUUIATION. 


21 


position  f,  and  then  it  continued  descending  on  the  (j{)posite  side  till  the  end  of 
y^  of  a  second  and  so  on.  We  can  also  easily  determine  where  the  vibrating 
point  was  to  be  found  at  the  end  of  any  fraction  of  this  hundred-and-twentieth  of 
a  second.  A  drawing  of  this  kind  consequently  shows  immediately  at  what  point  of 
its  path  a  vibrating  particle  is  to  be  found  at  any  given  instant,  and  hence  gives  a 
complete  image  of  its  motion.  If  the  reader  wishes  to  reproduce  the  motion  of  the 
vibrating  point,  he  has  only  to  cut  a  narrow  vertical  slit  in  a  piece  of  paper,  and 
place  it  over  fig.  6  or  fig.  7,  so  as  to  show  a  vei-y  small  portion  of  the  curve  through 
the  vertical  slit,  and  draw  the  book  slowly  but  uniformly  under  the  slit,  from  right 
to  left ;  the  white  or  black  point  in  the  slit  will  then  appear  to  move  backwards  and 
forwards  in  precisely  the  same  manner  as  the  original  drawing  point  attached  to 
the  fork,  only  of  course  much  more  slowly. 

We   are   not  yet   able    to   make  all   vibrating  bodies  describe  their  vibrations 


U 


directly    on    paper,    although    nmch 
methods  required    for   this    purpose. 


has    recently    been    made    in  the 
are  able  ourselves   to  draw   such 


progress 
But  we 
curves  for  all  sounding  bodies,  when  the  law  of  their  motion  is  known,  that  is, 
when  we  know  how  far  the  vibrating  point  will  be  from  its  mean  position  at  a,ny 
given  moment  of  time.  We  then  set  off  on  a  horizontal  line,  such  as  e  f,  fig.  7, 
lengths  corresponding  to  the  interval  of  time,  and  let  fall  perpendiculars  to  it  on^ 
either  side,  making  their  lengths  equal  or  proportional  to  the  distance  of  the  vibrat- 
ing point  from  its  mean  position,  and  then  by  joining  the  extremities  of  these  per- 
pendiculars we  obtain  a  curve  such  as  the  vibrating  body  would  have  drawn  if  it 
had  been  possible  to  make  it  do  so. 

Thus  fig.  8  represents  the  motion  of  the  hammer  raised  by  a  water-wheel,  or  of 
a  point  in  a  string  excited  by  a  violin  bow.  For  the  first  9  intervals  it  rises  slowly 
and  xuiiformly,  and  during  the  10th  it  falls  suddenly  down. 


LiA. 


V 


Fig.  9  represents  the  motion  of  the  ball  which  is  struck  up  again  as  soon  as  it^ 
"comes  down.     Ascent  and  descent  are  performed  with  equal  rapidity,  whereas  in 
fig.  8  the  ascent  takes  much  longer  time.    But  at  the  lowest  point  the  blow  suddenly 
changes  the  kind  of  motion. 

Physicists,  then,  having  in  their  mind  such  curvilinear  forms,  representing  the 
law  of  the  motion  of  sounding  bodies,  speak  briefly  of  the  form  of  vibratum  of  a 
sounding  body,  and  assert  that  the  (juality  of  tone  depends  on  the  form  of  vibration. 
This  assertion,  which  has  hitherto  been  based  simply  on  the  fact  of  our  knowing 
that  the  quality  of  the  tone  could  not  possibly  depenfl  on  the  periodic  time  of  a 
vibration,  or  on  its  amplitude  (p.  10c),  will  be  strictly  examined  hereafter.  It 
will  be  shown  to  be  in  so  far  correct  that  every  different  quality  of  tone  recpiires  a 
difterent  form  of  vibration,  but  on  the  other  hand  it  will  also  appear  that  different 
forms  of  vibration  may  correspond  to  the  same  quality  of  tone. 

On  exactly  and  carefully  examining  the  effect  produced  on  the  ear  by  difterent 
forms  of  vibration,  as  for  example  that  in  fig.  8,  corresponding  nearly  to  a  violin 


22  COMPOUND  AND  PARTIAL  TONES.  part  i. 

string,  we  meet  with  a  strange  and  imexpected  phenomenon,  long  known  indeed  to 
individual  musicians  and  ph^'sicists,  but  commonly  regarded  as  a  mere  curiosity, 
its  generality  and  its  great  significance  for  all  matters  relating  to  musical  tones  not 
having  been  recognised.  The  ear  when  its  attention  has  been  properly'  directed  to 
the  effect  of  the  vibrations  which  strike  it,  does  not  hear  merely  that  one  musical 
tone  whose  pitch  is  determined  by  the  period  of  the  vibrations  in  the  manner 
already  explained,  but  in  addition  to  this  it  becomes  aware  of  a  whole  series  of 
higher  musical  tones,  which  we  will  call  the  harmonic  upper  partial  tones,  and 
sometimes  simply  the  iipper  jMrtials  of  the  whole  musical  tone  or  note,  in  contra- 
distinction to  the  fundamental  or  prime  x><irtial  tone  or  simply  the  prime,  as  it  may 
be  called,  which  is  the  lowest  and  generally  the  loudest  of  all  the  partial  tones  and 
by  the  pitch  of  which  we  judge  of  the  pitch  of  the  whole  compound  musical  tone 
itself.  The  series  of  these  upper  partial  tones  is  precisely  the  same  for  all  com- 
H  pound  musical  tones  which  correspond  to  a  uniformly  periodical  motion  of  the  air. 
It  is  as  follows  : — 

The  first  upper  partial  tone  [or  second  partial  tone]  is  the  upper  Octave  of  the 
prime  tone,  and  makes  double  the  number  of  vibrations  in  the  same  time.  If  we 
call  the  prime  6',  this  upper  Octave  will  be  c. 

The  .second  upper  partial  tone  [or  third  partial  tone]  is  the  Fifth  of  this  Octave, 
or  g,  making  three  times  as  mtxny  vibrations  in  the  same  time  as  the  prime. 

The  third  upper  partial  tone  [or  fourth  partial  tone]  is  the  second  higher  Octave 
or  c',  making  four  times  as  many  vibrations  as  the  prime  in  the  same  time. 

The  fourth  upper  partial  tone  [or  fifth  partial  tone]  is  the  major  Third  of  this 
second  higher  Octave,  or  e ,  with  five  times  as  many  vibrations  as  the  prime  in  the 
same  time. 

The  fifth  upper  partial  tone  [or  sixth  partial  tone]  is  the  Fifth  of  the  second 
higher  Octave,  or  (f ,  making  six  times  as  many  vibrations  as  the  prime  in  the 
^  same  time. 

And  thus  they  go  on,  becoming  continually  fainter,  to  tones  making  7,  8,  9, 
ifcc,  times  as  many  vibrations  in  the  same  time,  -.s  the  prime  tone.  Or  in  musical 
notation 


i^^§=^=bEL=l? 


fj 


fj      "/>'[?     c"   d'   e"   ^y   fi"  ^^'a"  ~h"\)   h"    c" 


Ordinal  mmher  of  1       2       3  4       5       6  7        8     9     10     11       12     13       14      15      16 

Pitch  mfmber     66    132  198         264  380  396       462  528  594  660  726     792  858     924    990    1054* 

where  the  figures  [in  the  first  line]  beneath  show  how  many  times  the  corresponding 

pitch  number  is  greater  than  that  of  the  prime  tone  [and,  taking  the  lowest  note 

to  have  66  vibrations,  those  in  the  second  line  give  the  pitch  numbers  of  all  the 

*\  other  notes]. 

The  whole   sensation   excited   in  the  ear  by  a  periodic  vibration  of  the  air  we 

*  [This  diagram  has  been  slightly  altered  to  This  slightly  flattens  each  note,  and  slow  beats 
introduce  all  the  first  16  harmonic  partials  can  be  produced  in  ever_v  case  (except,  of 
of  C  66  (which,  excepting  11  and  13,  are  course,  11  and  13,  which  are  not  on  the 
given  on  the  Harmouical  as  harmonic  notes),  instrument)  up  to  16.  It  should  also  be  ob- 
aiid  to  show  the  notation,  symbolising,  both  in  served  that  the  pitch  of  the  beat  is  very  nearly 
letters  and  on  the  staff,  the  7th,  11th,  and  that  of  the  upper  {not  the  lower)  note  in  each 
13th  harmonic  partials,  which  are  not  used  in  case.  The  whole  of  these  16  harmonics  of  C  66 
general  music.  It  is  easy  to  show  on  the  (except  the  11th  and  13th)  can  Ije  played 
Harmonical  that  its  lowest  note,  C  of  this  at  once  on  the  Harmonical  by  means  of  the 
series,  contains  all  these  partials,  after  the  harmonical  bar,  first  without  and  then  with 
theory  of  the  beats  of  a  disturbed  unison  the  7th  and  14th.  The  whole  series  will  be 
has  been  explained  in  Chap.  VIII.  Keep  found  to  sound  like  a  single  fine  note,  and  the 
down  the  note  C,  and  touch  in  succession  the  7th  and  14th  to  materially  increase  its  rich- 
notes  c,  g,  c',  c',  cj',  &c.,  but  in  touching  the  latter  ness.  The  relations  of  the  partials  in  this  case 
press  the  fmger-key  such  a  little  way  down  may  be  studied  from  the  tables  in  the  footnotes 
that  the  tone  of  the  note  is  only  just  audible.  to  Chap.  X. —  Translator.'] 


DEFINITION  OF  TERMS  EMPLOYED. 


23 


have  called  a  mHsica/  tone.  We  now  find  that  this  is  ronijjoujtd,  c()ntainin<>-  a 
series  of  ditt'erent  tones,  which  we  distinguish  as  the  constituents  or  2M)-tial  tones 
of  the  compound.  The  first  of  these  constitnents  is  the  pritne  j^artial  tone  of  the 
compoiuid,  and  the  rest  its  harmonic  upper  partial  tones.  The  number  which 
shows  the  order  of  any  partial  tone  in  the  series  shows  how  many  times  its 
vibrational  number  exceeds  that  of  the  prime  tone.*  Thus,  the  second  partial 
tone  makes  twice  as  many,  the  third  three  times  as  many  vibrations  in  the  same 
time  as  the  prime  tone,  and  so  on. 

G.  S.  Ohm  was  the  first  to  declare  that  there  is  only  one  form  of  vibration 
which  will  give  rise  to  no  harmonic  upper  partial  tones,  and  which  will  therefore 
consist  solely  of  the  prime  tone.  This  is  the  form  of  vibration  which  we  have 
described  above  as  pecidiar  to  the  pendulum  and  tmiing-forks,  and  drawn  in  figs.  G 
and  7  (p.  10).  We  will  call  these  j^^ndular  vibrations,  or,  since  they  caiuiot  be 
analysed  into  a  compound  of  diflferent  tones,  simple  vibrations.  In  what  sense  not  H 
merely  other  musical  tones,  but  all  other  forms  of  vibration,  may  be  considered 
as  compowid,  will  be  shown  hereafter  (Chap.  IV.).  The  terms  simple  or  pendular 
vibration, f  will  therefore  be  used  as  synonymous.  We  have  hitherto  used  the 
expression  tone  and  musical  tone  indifferently.  It  is  absolutely  necessary  to  dis- 
tinguish in  acoustics  first,  a  musical  tone,  that  is,  the  impression  made  by  an// 
periodical  vibration  of  the  air ;  secondW,  a  simple  tone,  that  is,  the  impression 
l)roduced  by  a  simpde  or  pendular  vibration  of  the  air ;  and  thirdly  a  comjwund 
tone,  that  is,  the  impression  produced  by  the  simultaneous  action  of  several  simple 
tones  with  certain  definite  ratios  of  pitch  as  already  explained.  A  musical  tone 
may  be  either  simjde  or  comptound.     For  the  sake  of  brevity,  tone  will  be  used  in 


*  [The  ordinal  number  of  a  partial  tone 
in  general,  must  be  distinguished  from  the 
ordinal  number  of  an  upper  partial  tone  in 
particular.  For  the  same  tone  the  former 
number  is  always  greater  by  unity  than  the 
latter,  because  the  partials  in  general  include 
the  prime,  which  is  reckoned  as  the  first,  and 
the  upper  partials  exclude  the  prime,  which 
being  the  loudest  partial  is  of  course  not  an 
upper  partial  at  all.  Thus  the  partials  gene- 
rally numbered  2  3  4  5  6  7  8  9  are  the 
same  as  the  upper  partials  numbered  12  3 
4  5  6  7  8  respectively.  As  even  the 
Author  has  occasionally  failed  to  carry  out 
this  distinction  in  the  original  German  text, 
and  other  writers  have  constantly  neglected  it, 
too  much  weight  cannot  be  here  laid  upon  it. 
The  presence  or  absence  of  the  word  wppcr 
before  the  word  partml  must  always  be  care- 


fully observed.  It  is  safer  never  to  speak  of 
an  vipper  partial  by  its  ordinal  number,  but  to 
call  the  pfth  upixr  partial  the  sixth  partial, 
omitting  the  word  uyper  and  increasing  the  51 
ordinal  number  by  one  place.  And  so  in 
other  cases.  —  Translator.'] 

t  The  law  of  these  vibrations  may  be 
popularly  explained  by  means  of  the  constritc- 
tion  in  fig.  10.  Suppose  a  point  to  describe 
the  circle  of  which  c  is  the  centre  with  a 
uniform  velocity,  and  that  an  observer  stands 
at  a  considerable  distance  in  the  prolongation 
of  the  line  e  h,  so  that  he  does  not  see  the 
surface  of  the  circle  but  only  its  edge,  in 
which  case  the  point  will  appear  merely  to 
move  up  and  down  along  its  diameter  a  b. 
This  up  and  down  motion  would  take  place 
exactl}-  according  to  the  law  of  pendular 
vibration.     To  represent  this  motion  graphi- 


cally by  means  of  a  curve,  divide  the  length 
e  g,  supposed  to  correspond  to  the  time  of  a 
single  period,  into  as  many  (here  12)  equal 
parts  as  the  circumference  of  the  circle,  and 
draw  the  perpendiculars  1,  2,  3,  &c.,  on  the 
dividing  points  of  the  line  e  g,  in  order,  equal 
in  length  to  and  in  the  same  direction  with, 
those  drawn  in  the  circle  from  the  correspond- 
ing points  1,  2,  3,  &c.  In  this  way  we  obtain 
the  curve  drawn  in  fig.  10,  which  agrees   in 


form  witli  that  drawn  by  the  tuning-fork, 
fig.  6,  p.  206,  but  is  of  a  larger  size.  Mathe- 
matically expressed,  the  distance  of  the  vibrat- 
ing point  from  its  mean  position  at  any  time 
is  equal  to  the  sine  of  an  arc  proportional  to 
the  corresponding  time,  and  hence  the  form  of 
simple  vibrations  are  also  called  the  sivc- 
vibrations  [and  the  above  curve  is  also  known 
as  the  curve  of  sines']. 


24 


DEFINITION  OF  TERMS  EMPLOYED. 


the  general  sense  of  a  musical  tone,  leaving  the  context  or  a  prefixed  tjualitication 
to  determine  whether  it  is  simple  or  compound.  A  compound  tone  will  often  bo 
briefly  called  a  note,  and  a  simple  tone  will  also  be  frequently  called  a  imrtial,  when 
used  in  connection  with  a  compound  tone  :  otherwise,  the  full  expression  simple 
tone  will  be  employed.  A  note  has,  properly  speaking,  no  single  pitch,  as  it  is 
made  up  of  various  partials  each  of  which  has  its  own  pitch.  By  the  2'''^^'^^  of  a 
note  or  compo^ind  tone  then  we  shall  therefore  mean  the  ^)?YcA  of  its  lowest  pjartial 
or  prime  tone.  By  a  chord  or  combination  of  tones  we  mean  several  musical  tones 
(whether  simple  or  compound)  produced  by  difl^erent  instruments  or  diff'erent  parts 
of  the  same  instrument  so  as  to  be  heard  at  the  same  time.  •  The  facts  here  adduced 
show  us  then  that  every  musical  tone  in  which  harmonic  upper  partial  tones  can 
be  distinguished,  although  produced  by  a  single  instrument,  may  really  be  con- 
sidered as  in  itself  a  chord  or  combination  of  various  simple  tones.* 


H 


*  [The  above  paragraph  relating  to  the 
English  terms  used  in  this  translation,  neces- 
sarily differs  in  many  respects  from  the  original, 
in  which  a  justification  is  given  of  the  use 
made  by  the  Author  of  certain  German  ex- 
pressions. It  has  been  my  object  to  employ 
terms  which  should  be  thoroughly  English, 
and  should  not  in  any  way  recall  the  German 
words.  The  word  tone  in  English  is  extremely 
ambiguous.  Prof.  Tj'ndall  {Lectures  on  Sound, 
2nd  ed.  1869,  p.  117)  has  ventured  to  define  a 
tone  as  a  sini/ile  lone,  in  agreement  with  Prof. 
Helmholtz,  who  in  the  present  passage  limits 
the  German  word  Ton  in  the  same  way.  But 
I  felt  that  an  English  reader  could  not  be 
safely  trusted  to  keep  this  very  peculiar  and 
important  class  of  musical  tones,  which  he 
has  very  rarely  or  never  heard  separately, 
invariably  distinct  from  those  musical  tones 

•fl  with  which  he  is  familiar,  unless  the  word 
tone  were  uniformly  qualified  by  the  epithet 
simple.  The  only  exception  I  could  make  was 
in  the  case  of  a  partial  tone,  which  is  received 
at  once  as  a  new  conception.  Even  Prof. 
Helmholtz  himself  has  not  succeeded  in  using 
his  word  Ton  consistently  for  a  simple  tone 
only,  and  this  was  an  additional  warning  to 
me.  English  musicians  have  been  also  in 
the  habit  of  using  tone  to  signify  a  certain 
musical  interval,  and  semitone  for  half  of  that 
interval,  on  the  equally  tempered  scale.  In 
this  case  I  write  Tone  and  Semitone  with 
capital  initials,  a  practice  which,  as  already 
explained  (note,  p.  13c?'),  I  have  found  con- 
venient for  the  names  of  all  intervals,  as 
Thirds,  Fifths,  &c.  Prof.  Helmholtz  uses  the 
word  Klang  for  a  musical  tone,  which  gene- 

^  rally,  but  not  always,  means  a  compound  tone. 
Prof.  Tyndall  (ibid.)  therefore  proposes  to  use 
the  English  word  clang  in  the  same  sense. 
But  clang  has  already  a  meaning  in  English, 
thus  defined  by  Webster :  '  a  sharp  shrill 
sound,  made  by  striking  together  metallic 
substances,  or  sonorous  bodies,  as  the  clang 
of  arms,  or  any  like  sound,  as  the  claiig  of 
trumpets.  This  word  implies  a  degree  of 
harshness  in  the  sound,  or  more  harshness 
than  clink.'  Interpreted  scientifically,  then, 
clang  according  to  this  definition,  is  either 
noise  or  one  of  those  musical  tones  until  in- 
harmonic upper  partials,  which  will  be  sub- 
sequently explained.  It  is  therefore  totally 
unadapted  to  represent  a  musical  tone  in 
general,  for  which  the  simple  word  tone  seems 
eminently  suited,  being  of  course  originally 
the  tone  produced  by  a  stretched  string.  The 
coiomon   word   note,   properly   the   mark    by 


which  a  musical  tone  is  written,  will  also,  in 
accordance  with  the  general  practice  of  musi- 
cians, be  used  for  a  musical  tone,  which  is 
generally  compound,  without  necessarily  im- 
plying that  it  is  one  of  the  few  recognised 
tones  in  our  musical  scale.  Of  course,  if 
clang  could  not  be  used.  Prof.  Tyndall's 
suggestion  to  translate  Prof.  Helmholtz's 
Klangfarbc  by  clangtint  (ibid.)  fell  to  the 
ground.  I  can  find  no  valid  reason  for  sup- 
planting the  time-honoured  expression  qualitg 
of  tone.  Prof.  Tyndall  [ibid.)  quotes  Dr. 
Young  to  the  effect  that '  this  quality  of  sound 
is  sometimes  called  its  register,  colour,  or 
timbre'.  Register  has  a  distinct  meaning  in 
vocal  music  which  must  not  be  disturbed. 
Timbre,  properly  a  kettledrum,  then  a  helmet, 
then  the  coat  of  arms  surmounted  with  a 
helmet,  then  the  official  stamp  bearing  that 
coat  of  arms  (now  used  in  France  for  a 
postage  label),  and  then  the  mark  which 
declared  a  thing  to  be  what  it  pretends  to  be, 
Burns's  'guinea's  stamp,'  is  a  foreign  word, 
often  odiously  mispronounced,  and  not  worth 
preserving.  Colour  I  have  never  met  with 
as  applied  to  music,  except  at  most  as  a 
passing  metaphorical  expression.  But  the 
difference  of  tones  in  qualitg  is  familiar  to 
our  language.  Then  as  to  the  Partial  Tones, 
Prof.  Helmholtz  uses  Theiltone  and  Particd- 
tone,  which  are  aptly  Englished  by  partial 
simple  tones.  The  words  simple  and  tone, 
however,  may  be  omitted  when  partials  is 
employed,  as  partials  are  necessarily  both 
tones  and  simple.  The  constituent  tones  of  a 
chord  may  be  either  simple  or  compound. 
The  Grundion  or  fundamental  tone  of  a 
compound  tone  then  becomes  its  prime  tone, 
or  briefly  its  prime.  The  Grundton  or  root  of 
a  chord  will  be  further  explained  hereafter. 
Upper  partial  (simple)  tones,  that  is,  the 
partials  exclusive  of  the  prime,  even  when 
harmonic  (that  is,  for  the  most  part,  belong- 
ing to  the  first  six  partial  tones),  must  be 
distinguished  from  the  sounds  usually  called 
harmonics  when  produced  on  a  violin  or  harp 
for  instance,  for  such  harmonics  are  not  neces- 
sarily simple  tones,  but  are  more  generally 
compounds  of  some  of  the  complete  series  of 
j)artial  tones  belonging  to  the  musical  tone  of 
the  whole  string,  selected  by  damping  the 
remainder.  The  fading  harmonics  heard  in 
listening  to  the  sound  of  a  pianoforte  string, 
struck  and  undamped,  as  the  sound  dies  away, 
are  also  compound  and  not  simple  partial 
tones,  but  as  they  have  the  successive  partials 
for   their   successive   primes,    they   have   the 


CHAPS.  I.  11.    COEXISTENCE  OF  DISTINCT  WAVES  OF  SOUND.  25 

Now,  since  quality  of  tone,  as  we  have  seen,  depends  on  the  form  of  vibration, 
which  also  determines  the  occurrence  of  upper  partial  tones,  we  have  to  inquire 
how  far  differences  in  quality  of  tone  depend  on  different  force  or  loudness  of  upper 
partials.  This  inq\iiry  will  be  found  to  give  a  means  of  clearing-  up  our  concep- 
tions of  what  has  liitherto  been  a  perfect  enigma, — the  nature  of  quality  of  tone. 
And  we  must  then,  of  course,  attempt  to  explain  how  the  ear  manages  to  analyse 
every  musical  tone  into  a  series  of  partial  tones,  and  what  is  the  meaning  of  this 
analysis.     These  investigations  will  engage  our  attention  in  the  following  chapters. 


CHAPTER  II. 

ON     THE    COMPOSITION    OF    VIBRATIONS. 


A.T  the  end  of  the  last  chapter  we  came  upon  the  remarkable  fact  that  the  human 
ear  is  capable,  under  certain  conditions,  of  separating  the  musical  tone  produced 
by  a  single  musical  instrument,  into  a  series  of  simple  tones,  namely,  the  prime 
partial  tone,  and  the  various  upper  partial  tones,  each  of  which  produces  its  own 
separate  sensation.  That  the  ear  is  capable  of  distinguishing  from  each  other 
tones  proceeding  from  different  sources,  that  is,  which  do  not  arise  from  one  and 
the  same  sonorous  body,  we  know  from  daily  experience.  There  is  no  difficulty 
during  a  concert  in  following  the  melodic  progression  of  each  individual  instru- 
ment or  voice,  if  we  direct  our  attention  to  it  exclusively ;  and,  after  some  practice, 
most  persons  can  succeed  in  following  the  simultaneous  progression  of  several 
united  parts.  This  is  true,  indeed,  not  merely  for  musical  tones,  but  also  for 
noises,  and  for  mixtures  of  music  and  noise.  When  several  persons  are  speaking 
at  once,  we  can  generally  listen  at  pleasure  to  the  words  of  any  single  one  of  them,  H 
and  even  understand  those  words,  provided  that  they  are  not  too  much  overpowered 
by  the  mere  loudness  of  the  others.  Hence  it  follows,  first,  that  many  different 
trains  of  waves  of  sound  can  be  propagated  at  the  same  time  through  the  same 
mass  of  ail-,  without  mutual  disturbance  ;  and,  secondly,  that  the  human  ear  is 
capable  of  again  analysing  into  its  constituent  elements  that  composite  motion  of 
the  air  which  is  produced  by  the  simultaneous  action  of  several  musical  instru- 
ments. We  will  first  investigate  the  nature  of  the  motion  of  the  air  when  it  is 
produced  by  several  simultaneous  musical  tones,  and  how  such  a  compound  motion 
is  distinguished  from  that  due  to  a  single  musical  tone.  We  shall  see  that  the  ear 
has  no  decisive  test  by  which  it  can  in  all  cases  distinguish  between  the  effect  of  a 

pitch  of  those  partials.  But  these  fading  meaning  upper,  but  the  English  preposition 
harmonics  are  not  regular  compound  tones  of  over  is  equivalent  to  the  German  preposition 
the  kind  described  on  p.  22«,  because  the  lower  iiher.  Compare  Obcrzolm,  &n  'upper  tooth,' ^ 
partials  are  absent  one  after  another.  Both  i.e.,  a  tooth  in  the  upper  jaw,  with  UeherzaJm, 
sets  of  harmonics  serve  to  indicate  the  exist-  an  '  overtooth,'  i.e.,  one  grown  over  another, 
ence  and  place  of  the  partials.  But  they  are  a  projecting  tooth.  The  continual  recurrence 
no  more  those  upper  partial  tones  themselves,  of  such  words  as  cJancj,  clancjtint,  overtone, 
than  the  original  compound  tone  of  the  string  would  combine  to  give  a  strange  un-English 
is  its  own  prime.  Great  confusion  of  thought  appearance  to  a  translation  from  the  German, 
having,  to  my  own  knowledge,  arisen  from  On  the  contrary  I  have  endeavoured  to  put  it 
conionndiug  such  ha rmunics  with  tqjpcr  parti(il  into  as  straightforward  EngHsh  as  possible. 
tones,  I  have  generally  avoided  using  the  am-  But  for  those  acquainted  with  the  original  and 
biguous  substantive  Af/r/iwy/uV.  Properly  speak-  with  Prof.  Tyndall's  work,  this  explanation 
ing  the  harmonics  of  any  compound  tone  are  seemed  necessary.  Finally  I  would  caution 
other  compound  tones  of  which  the  primes  are  the  reader  against  using  overtones  for  partial 
partials  of  the  original  compound  tone  of  tones  in  general,  as  almost  every  one  who 
which  they  are  said  to  be  harmonics.  Prof.  adopts  Prof.  Tyndall's  word  is  in  the  habit  of 
Helmholtz's  term  Oherfihie  is  merely  a  con-  doing.  Indeed  I  have  in  the  course  of  this 
traction  for  Oberpartiattonc,  but  the  casual  translation  observed,  that  even  Prof.  Helmholtz 
resemblance  of  the  sounds  of  ober  and  over,  has  himself  has  been  occasionally  misled  to  em- 
led  Prof.  Tyndall  to  the  erroneous  translation  ploy  Obertone  in  the  same  loos^e  manner.  See 
overtones.     The   German   ober  is  an  adjective  my  remarks  in  note,  p.  23f. —  Translator.] 


26  COMPOSITION  OF  WAVES.  part  i. 

motion  of  the  air  caused  by  several  different  musical  tones  arising  from  different 
sources,  and  that  caused  by  the  musical  tone  of  a  single  sounding  body.  Hence 
the  ear  has  to  analyse  the  composition  of  single  musical  tones,  under  proper  con- 
ditions, by  means  of  the  same  faculty  which  enabled  it  to  analyse  the  composition 
of  simultaneous  musical  tones.  We  shall  thus  obtain  a  clear  concei^tion  of  v.-hat 
is  meant  by  analysing  a  single  musical  tone  into  a  series  of  partial  simple  tones, 
and  we  shall  perceive  that  this  phenomenon  depends  upon  one  of  the  most 
essential  and  fundamental  properties  of  the  human  ear. 

We  begin  by  examining  the  motion  of  the  air  which  corresponds  to  several 
simple  tones  acting  at  the  same  time  on  the  same  mass  of  air.  To  illustrate  this 
kind  of  motion  it  will  be  again  convenient  to  refer  to  the  waves  foi-med  on  a  calm 
surface  of  water.  We  have  seen  (p.  9a)  that  if  a  point  of  the  surface  is  agitated  by  a 
stone  thrown  upon  it,  the  agitation  is  propagated  in  rings  of  waves  over  the  surface 

f  to  more  and  more  distant  points.  Now,  throw  two  stones  at  the  same  time  on  to 
different  points  of  the  surface,  thus  producing  two  centres  of  agitation.  Each  will 
give  rise  to  a  separate  ring  of  waves,  and  the  two  rings  gradually  expanding,  will 
finally  meet.  Where  the  waves  thus  come  together,  the  water  will  be  set  in 
motion  by  both  kinds  of  agitation  at  the  same  time,  but  this  in  no  wise  prevents 
botli  series  of  waves  from  advancing  further  over  the  surface,  just  as  if  each  were 
alone  present  and  the  other  had  no  existence  at  all.  As  they  proceed,  those 
parts  of  both  rings  which  had  just  coincided,  again  appear  separate  and  mialtered 
in  form.  These  little  waves,  caused  by  throwing  in  stones,  may  be  accompanied 
by  other  kinds  of  waves,  such  as  those  due  to  the  wind  or  a  passing  steamboat., 
Our  circles  of  waves  will  spread  out  over  the  water  tluis  agitated,  with  the  same 
quiet  regularity  as  they  did  upon  the  calm  surface.  Neither  will  the  greater  waves 
be  essentially  disturbed  by  the  less,  nor  the  less  by  the  greater,  provided  the  waves 
never  break  ;  if  that  happened,  their  regular  course  would  certainly  be  impeded. 

H  Indeed  it  is  seldom  possible  to  survey  a  large  surface  of  water  from  a  high 
point  of  sight,  without  perceiving  a  great  multitude  of  different  systems  of  waves 
mutually  overtopping  and  crossing  each  other.  This  is  best  seen  on  the  surface  of 
the  sea,  viewed  from  a  lofty  cliff,  when  there  is  a  lull  after  a  stiff  breeze.  We  first 
see  the  great  waves,  advancing  in  far-stretching  ranks  from  the  blue  distance,  here 
and  there  more  clearly  marked  oiit  by  their  white  foaming  crests,  and  following 
one  another  at  regular  intervals  towards  the  shore.  From  the  shore  they  rebound, 
in  different  directions  according  to  its  sinuosities,  and  cut  obliquely  across  the 
advancing  waves.  A  passing  steamboat  forms  its  own  wedge-shaped  wake  of 
waves,  or  a  bird,  dai'ting  on  a  fish,  excites  a  small  circular  system.  The  eye  of  the 
spectator  is  easily  able  to  pursue  each  one  of  these  diHerent  trains  of  waves,  great 
and  small,  wide  and  narrow,  straight  and  curved,  and  observe  how  each  passes 
over  the  surface,  as  undisturbedly  as  if  the  water  over  which  it  flits  Avere  not 
agitated  at  the  same  time  by  other  motions  and  other  forces.     I  must  own  that 

II  whenever  I  attentively  observe  this  spectacle  it  awakens  in  me  a  peculiar  kind  of 
intellectual  pleasure,  because  it  bares  to  the  bodily  eye,  what  the  mind's  eye  grasps 
only  by  the  help  of  a  long  series  of  complicated  conclusions  for  the  waves  of  the 
invisible  atmospheric  ocean. 

We  have  to  imagine  a  perfectly  similar  spectacle  proceeding  in  the  interior  of  a 
ball-room,  for  instance.  Hera  we  have  a  number  of  musical  instruments  in  action, 
speaking  men  and  women,  rustling  garments,  gliding  feet,  clinking  glasses,  and  so 
on.  All  these  causes  give  rise  to  systems  of  waves,  which  dart  through  the  mass 
of  air  in  the  room,  are  reflected  from  its  walls,  return,  strike  the  opposite  wall,  are 
again  reflected,  and  so  on  till  they  die  out.  We  have  to  imagine  that  from  the 
mouths  of  men  and  from  the  deeper  musical  instruments  there  proceed  waves  of 
from  8  to  12  feet  in  length  [c  to  F],  from  the  lips  of  the  women  waves  of  2  to  4 
feet  in  length  [c"  to  c'],  from  the  rustling  of  the  dresses  a  fine  small  crumple  of 
wave,  and  so  on ;  in  short,  a  tumbled  entanglement  of  the  most  difterent  kinds  of 
motion,  complicated  beyond  conception. 


CHAP.  II.  ALGEBRAICAL  ADDITIOX  OF  WAVES.  27 

And  yet,  ;is  the  ear  is  able  to  distinguish  all  the  sei)arate  constituent  parts  of 
this  confused  whole,  we  are  forced  to  conclude  that  all  these  different  systems  of 
wave  coexist  in  the  mass  of  air,  and  leave  one  another  mntually  undisturbed. 
But  how  is  it  possible  for  them  to  coexist,  since  every  individual  train  of  waves  has 
at  any  particular  point  in  the  mass  of  air  its  own  particular  degree  of  condensa- 
tion and  rarefaction,  which  determines  the  velocity'  of  the  particles  of  air  to  this 
side  or  that  ?  It  is  evident  that  at  each  point  in  the  mass  of  air,  at  each  instant 
of  time,  there  can  be  only  one  single  degree  of  condensation,  and  that  the  particles 
of  air  can  be  moving  with  only  one  single  determinate  kind  of  motion,  having  only 
one  single  determinate  amount  of  velocity,  and  passing  in  only  one  single  deter- 
minate direction. 

What  happens  under  such  circumstances  is  seen  directly  by  the  eye  in  the 
waves  of  water.  If  where  the  water  shows  large  waves  we  throw  a  stone  in,  tiie 
waves  thus  caused  will,  so  to  speak,  cut  into  the  larger  moving  surface,  and  thislj 
surface  will  be  partly  raised,  and  partlj-  depressed,  by  the  new  waves,  in  such  a 
way  that  the  fresh  crests  of  the  rings  will  rise  just  as  much  above,  and  the  troughs 
sink  just  as  much  below  the  curved  surfaces  of  the  previous  larger  waves,  as  they 
would  have  risen  above  or  svink  below  the  horizontal  surface  of  calm  water. 
Hence  where  a  crest  of  the  smaller  system  of  rings  of  waves  comes  upon  a  crest 
of  the  greater  system  of  waves,  the  surface  of  the  water  is  raised  by  the  sum  of 
the  two  heights,  and  where  a  trough  of  the  former  coincides  with  a  trough  of  the 
latter,  the  surface  is  depressed  by  the  sum  of  the  two  depths.  This  may  be 
expressed  more  briefly  if  we  consider  the  heights  of  the  crests  above  the  level  of 
the  surface  at  rest,  as  positive  magnitudes,  and  the  depths  of  the  troughs  as  negative 
magnitudes,  and  then  form  the  so-called  algebraical  sum  of  these  positive  and 
negative  magnitudes,  in  which  case,  as  is  well  known,  two  positive  magnitudes 
(heights  of  crests)  must  be  added,  and  similarly  for  two  negative  magnitudes  (depths 
of  troughs) ;  but  when  both  negative  and  positive  concur,  one  is  to  be  subtracted  U 
from  the  other.  Performing  the  addition  then  in  this  algebraical  sense,  we  can 
express  our  description  of  the  surface  of  the  water  on  which  two  systems  of  waves 
concur,  in  the  following  simple  manner :  The  distance  of  the  surface  of  the  water 
at  any  point  from  its  jjosition  of  rest  is  at  any  moment  eqiial  to  the  [alyeljraica/] 
sum  of  the  distances  at  vjliich  it  ^vould  have  stood  had  each  wave  acted  separately 
at  the  same  jjlace  and  at  the  same  time. 

The  eye  most  clearly  and  easily  distinguishes  the  action  in  such  a  case  as  has 
been  just  adduced,  where  a  smaller  circular  system  of  waves  is  produced  on  a  large 
rectilinear  system,  because  the  two  systems  are  then  strongly  distinguished  from 
each  other  both  by  the  height  and  shape  of  the  waves.  But  with  a  little  attention 
the  eye  recognises  the  same  fact  even  when  the  two  systems  of  waves  have  but 
slightly  diff"erent  forms,  as  when,  for  example,  long  rectilinear  waves  advancing 
towards  the  shore  concur  with  those  reflected  from  it  in  a  slightly  different 
direction.  In  this  case  we  observe  those  well-known  comb-backed  waves  where  H 
the  crest  of  one  system  of  waves  is  heightened  at  some  points  by  the  crests  of  the 
other  system,  and  at  others  depressed  by  its  troughs.  The  multiplicity  of  forms 
is  here  extremely  great,  and  any  attempt  to  describe  them  would  lead  us  too 
far.  The  attentive  observer  will  readily  comprehend  the  result  by  examining 
any  disturbed  surface  of  water,  without  further  description.  It  will  s\iffice  for  our 
purpose  if  the  first  example  has  given  the  reader  a  clear  conception  of  what  is 
meant  by  adding  waves  together/'' 

Hence  although  the  surface  of  the  water  at  any  instant  of  time  can  assume 
only  one  single  form,  while  each  of  two  different  systems  of  waves  simultaneously 
attempts  to  impress  its  own  shape  upon  it,  we  are  able  to  suppose  in  the  above 

*  Tho  velocities  and  displacements  of  the  addition  of  waves  as  is  spoken  of  in  the  text, 

particles  of  water  are  also  to  be  added  accord-  is  not  perfectly  correct,  unless  the  heights  of 

ing  to  the  law  of  the  so-called  parallelogram  the  waves  are  infinitely  small  in  comparison 

of  forces.      Strictly  speaking,  such  a  simple  with  their  lengths. 


28  ALGEBRAICAL  ADDITION  OF  WAVES.  part  i. 

sense  that  the  two  systems  coexist  and  are  superimposed,  by  considering  the 
actual  elevations  and  depressions  of  the  surface  to  be  suitably  separated  into  two 
parts,  each  of  which  belongs  to  one  of  the  systems  alone. 

In  the  same  sense,  then,  there  is  also  a  superimposition  of  different  systems  of 
sound  in  the  air.  By  each  train  of  waves  of  sound,  tlie  density  of  the  air  and  the 
velocity  and  position  of  the  particles  of  air,  are  temporarily  altered.  There  are 
places  in  the  wave  of  sound  comparable  with  the  crests  of  the  waves  of  water,  in 
which  the  quantity  of  the  air  is  increased,  and  the  air,  not  having  free  space  to 
escape,  is  condensed ;  and  other  places  in  the  mass  of  air,  comparable  to  the 
troughs  of  the  waves  of  water,  having  a  diminished  quantity  of  air,  and  hence 
diminished  density.  It  is  true  that  two  different  degrees  of  density,  produced  by 
two  different  systems  of  waves,  cannot  coexist  in  the  same  place  at  the  same  time ; 
nevertheless  the  condensations  and  rarefactions  of  the   air  can   be   (algebraically) 

H  added,  exactly  as  the  elevations  and  depressions  of  the  surface  of  the  water  in  the 
former  case.  Where  two  condensations  are  added  we  obtain  increased  condensation, 
where  two  rarefactions  are  added  we  have  increased  rarefaction  ;  while  a  concur- 
rence of  condensation  and  rarefaction  mutually,  in  whole  or  in  part,  destroy  or 
neutralise  each  other. 

The  displacements  of  the  particles  of  air  are  compounded  in  a  similar  manner. 
If  the  displacements  of  two  different  systems  of  waves  are  not  in  the  same  direc- 
tion, they  are  compounded  diagonally  ;  for  example,  if  one  system  would  drive  a 
particle  of  air  upwards,  and  another  to  the  right,  its  real  path  will  be  obliquely 
upwards  towards  the  right.  For  our  present  purpose  there  is  no  occasion  to  enter 
more  particularly  into  such  compositions  of  motion  in  different  directions.  We 
are  only  interested  in  the  effect  of  the  mass  of  air  upon  the  ear,  and  for  this  we 
are  only  concerned  with  the  motion  of  the  air  in  the  passages  of  the  ear.  Now  the 
passages  of  our  ear  are  so  narrow  in  comparison  with  the  length  of  the  waves  of 

^  sound,  that  we  need  only  consider  such  motions  of  the  air  as  are  parallel  to  the 
axis  of  the  passages,  and  hence  have  only  to  distinguish  displacements  of  the 
particles  of  air  outwards  and  inwards,  that  is  towards  the  outer  air  and  towards 
the  interior  of  the  ear.  For  the  magnitude  of  these  displacements  as  well  as  for 
their  velocities  with  which  the  particles  of  air  move  outwards  and  inwards,  the 
same  (algebraical)  addition  holds  good  as  for  the  crests  and  troughs  of  waves  of 
water. 

Hence,  vjhen  several  sonorous  bodies  in  the  surroxmding  atmosphere,  simnl- 
taneously  excite  different  systems  of  waves  of  sound,  the  changes  of  density  of  the 
air,  and  the  disj)lacements  and  velocities  of  the  ^)a^*^?'c^6's  of  the  air  ivithin  the 
passages  of  the  ear,  are  each  equal  to  the  [algebraical)  sum  of  the  corresponding 
changes  of  density,  disjolacements,  and-  velocities,  inhich  each  system  of  waves 
would  have  sejmrately  produced,  if  it  had  acted  independently  ;  *  and  in  this  sense 
we  can  say  that  all  the  separate  vibrations  which  separate  waves  of  sound  would 

H  have  produced,  coexist  undisturbed  at  the  same  time  within  the  passages  of  our  ear. 
After  having  thus  in  answer  to  the  first  question  explained  in  what  sense  it  is 
possible  for  several  different  systems  of  waves  to  coexist  on  the  same  surface  of 
water  or  within  the  same  mass  of  air,  we  proceed  to  determine  the  means  possessed 
by  our  organs  of  sense,  for  analysing  this  composite  whole  into  its  original  consti- 
tuents. 

I  have  already  observed  that  an  eye  which  surveys  an  extensive  and  disturbed 
surface  of  water,  easily  distinguishes  the  separate  systems  of  waves  from  each 
other  and  follows  their  motions.  The  eye  has  a  great  advantage  over  the  ear  in 
being  able  to  survey  a  large  extent  of  surface  at  the  same  moment.  Hence  the 
eye  readily  sees  whether  the  individual  waves  of  water  are  rectilinear  or  curved, 
and  whether  they  have  the  same  centre  of  curvature,  and  in  what  direction  they 

*  The  same  is  true  for  the  whole  mass  of       according  to  the  law  of  the  parallelogram  of 
external  air,  if  only  the  addition  of  the  dis-       forces, 
placements   in   different    dii'ections   is   made 


CHAP.  II.  EYE  AND   EAR  C'ONTRASTED.  29 

are  advancin"'.  All  these  observations  assist  it  in  determining  whotlier  two  systems 
of  waves  are  connected  or  not,  and  hence  in  discovering  their  corresponding  parts. 
Moreover,  («i  the  snrface  of  the  water,  waves  of  unequal  length  advance  with 
unecpial  velocities,  so  that  if  they  coincide  at  one  moment  to  such  a  degree  as  to 
be  difficult  to  distinguish,  at  the  next  instant  one  train  pushes  on  and  the  other 
lags  behind,  so  that  they  become  again  separately  visible.  In  this  way,  then,  the 
observer  is  greatly  assisted  in  referring  each  system  to  its  point  of  departure,  and 
in  keeping  it  distinctly  visible  during  its  further  course.  For  the  eye,  then,  two 
systems  of  waves  having  difterent  points  of  departure  can  never  coalesce ;  for 
example,  such  as  arise  from  two  stones  thrown  into  the  water  at  different  points. 
If  in  any  one  place  the  rings  of  wave  coincide  so  closely  as  not  to  be  easily 
separable,  they  always  remain  separate  during  the  greater  part  of  their  extent. 
Hence  the  eye  could  not  be  easily  brought  to  confuse  a  compound  with  a  simple 
undulatory  motion.  Yet  this  is  precisely  what  the  ear  does  under  similar  circum-H 
stances  when  it  separates  the  musical  tone  which  has  proceeded  from  a  single 
source  of  sound,  into  a  series  of  simple  partial  tones. 

But  the  ear  is  much  more  unfavourably  situated  in  relation  to  a  system  of  waves 
of  sound,  than  the  eye  for  a  system  of  waves  of  water.  The  ear  is  affected  only 
by  the  motion  of  that  mass  of  air  which  happens  to  be  in  the  immediate  neigh- 
bourhood of  its  tympanum  within  the  aural  passage.  Since  a  transverse  section 
of  the  aural  passage  is  comparatively  small  in  comparison  with  the  length  of  waves 
of  sound  (which  for  serviceable  musical  tones  varies  from  6  inches  to  .32  feet),*  it 
corresponds  to  a  single  point  of  the  mass  of  air  in  motion.  It  is  so  small  that 
distinctly  different  degrees  of  density  or  velocity  could  scarcely  occur  upon  it, 
because  the  positions  of  greatest  and  least  density,  of  greatest  positive  and  nega- 
tive velocity,  are  always  separated  by  half  the  length  of  a  wave.  The  ear  is 
therefore  in  nearly  the  same  condition  as  the  eye  would  be  if  it  looked  at  one  point 
of  the  surface  of  the  water,  through  a  long  narrow  tube,  which  would  permit  of  ^ 
seeing  its  rising  and  falling,  and  were  then  required  to  undertake  an  analysis 
of  the  compound  waves.  It  is  easily  seen  that  the  eye  would,  in  most  cases, 
completely  fail  in  the  solution  of  such  a  problem.  The  ear  is  not  in  a  condition 
to  discover  how  the  air  is  moving  at  distant  spots,  Avhether  the  waves  which  strike 
it  are  spherical  or  plane,  whether  they  interlock  in  one  or  more  circles,  or  in  what 
direction  they  are  advancing.  The  cii'cumstances  on  which  the  eye  chiefly  depends 
for  foi'ming  a  judgment,  are  all  absent  for  the  ear. 

If,  then,  notwithstanding  all  these  difficulties,  the  ear  is  capable  of  distin- 
guishing musical  tones  arising  from  different  sources — and  it  really  shows  a 
marvellous  readiness  in  so  doing— it  must  employ  means  and  possess  properties 
altogether  difterent  from  those  employed  or  possessed  by  the  eye.  But  whatever 
these  means  may  be — and  we  shall  endeavour  to  determine  them  hereafter — it 
is  clear  that  the  analysis  of  a  composite  mass  of  musical  tones  must  in  the  first 
place  be  closely  connected  with  some  determinate  properties  of  the  motion  of  the  ^ 
air,  capable  of  impressing  theniselves  even  on  such  a  very  minute  mass  of  air  as 
that  contained  in  the  aural  passage.  If  the  motions  of  the  particles  of  air  in  this 
passage  are  the  same  on  two  different  occasions,  the  ear  will  receive  the  same 
sensation,  whatever  be  the  origin  of  those  motions,  whether  they  spring  from  one 
or  several  sources. 

We  have  already  explained  that  the  mass  of  air  which  sets  the  tympanic 
membrane  of  the  ear  in  motion,  so  far  as  the  magnitudes  here  considered  are 
concerned,  must  be  looked  upon  as  a  single  point  in  the  surrounding  atmosphere. 
Are  there,  then,  any  peculiarities  in  the  motion  of  a  single  particle  of  air  which 
would  differ  for  a  single  musical  tone,  and  for  a  combination  of  musical  tones  ? 
We  have  seen  that  for  each  single  musical  tone  there  is  a  corresponding  periodical 

*  [These  are  of  course  rather  more  than  flue  organ  pipes.  See  Chap.  Y.  sect.  5,  and 
twice  the  length  of   the  corresponding   open       compare  p.  26rf. — Trmislator.] 


30 


COMPOSITION  OF  SIMPLE  WAVES. 


motion  of  the  air,  and  that  its  pitch  is  determined  by  the  length  of  the  periodic 
time,  but  that  the  kind  of  motion  during  any  one  single  period  is  perfectly  arbitrary, 
and  may  indeed  be  infinitely  various.  If  then  the  motion  of  the  air  lying  in  the 
aural  passage  is  not  periodic,  or  if  at  least  its  periodic  time  is  not  as  short  as  that 
of  an  audible  musical  tone,  this  fact  will  distinguish  it  from  any  motion  which 
belongs  to  a  musical  tone  ;  it  must  belong  either  to  noises  or  to  several  simultaneous 
musicll  tones.  Of  this  kind  are  really  the  greater  number  of  cases  where  the  dif- 
ferent musical  tones  have  been  only  accidentally  combined,  and  are  therefore  not 
designedly  framed  into  musical  chords;  nay,  even  where  orchestral  music  is  per- 
fornied,  the  method  of  tempered  tuning  which  at  present  prevails,  prevents  an 
accurate  fulfilment  of  the  conditions  under  which  alone  the  resulting  motion  of 
the  air  can  be  exactly  periodic.  Hence  in  the  greater  number  of  cases  a  want 
of  periodicity  in  the  motion  might  furnish  a  mark  for  distinguishing  the  presence 
^  of  a  composite  mass  of  musical  tones. 

But  a  composite  mass  of  musical  tones  may  also  give  rise  to  a  jmrely  periodic 
motion  of  the  air,  namely,  token  all  the  musical  tones  which  intermingle,  have 
pitch  numbers  which  are  all  multiples  of  one  and  the  same  old  mimher,  or  which 

Fio.  11. 


comes  to  the  same  thing,  when  all  these  musical  tones,  so  far  as  their  pjitch  is 
concerned,  may  he  regarded  as  the  upper  partial  tones  of  the  same  prime  tone.  It 
was  mentioned  in  Chapter  I.  (p.  22a,  h)  that  the  pitch  numbers  of  the  upper  partial 
tones  are  multiples  of  the  pitch  number  of  the  prime  tone.  The  meaning  of  this 
rule  will  be  clear  from  a  particular  example.  The  curve  A,  fig.  II,  represents  a 
pendular  motion  in  the  manner  explained  in  Chap.  I.  (p.  21/^),  as  produced  in  the 
air  of  the  aural  passage  by  a  tuning-fork  in  action.  The  horizontal  lengths  in  the 
curves  of  fig.  11,  consequently  represent  the  passing  time,  and  the  vertical  heights 
the  corresponding  displacements  of  the  particles  of  air  in  the  aural  passage.  Now 
suppose  that  \\ith  the  first  simple  tone  to  which  the  curve  A  corresponds,  there  is 
sounded  a  second  simple  tone,  represented  by  the  curve  B,  an  Octave  higher  than 
the  first.  This  condition  requires  that  two  vibrations  of  the  curve  B  should  be 
made  in  the  same  time  as  one  vibration  of  the  curve  A.  In  A,  the  sections  of  the 
curve  d„8  and  8  8i  are  perfectly  equal  and  similar.  The  curve  B  is  also  divided 
into  equal  and  similar  sections  e  e  and  c  ej  by  the  points  e,  c,  €,.  We  could  cer- 
tainly halve  each  of  the  sections  e  e  and  c  c„  and  thus  obtain  equal  and  similar 
sections,  each  of  which  would  then  correspond  to  a  single  period  of  B.     But  by 


CHAP.  II.  COMPOSITION   OF  .SIMPLE   WAVES.  31 

taking  sections  consisting  of  two  periods  of  B,  we  divide  B  into  larger  sections, 
each  of  which  is  of  the  same  horizontal  length,  and  hence  corres])onds  to  the  same 
duration  of  time,  as  the  sections  of  A. 

If,  then,  both  simple  tones  are  heard  at  once,  and  the  times  of  the  points  e  and 
dj,  €  and  8,  e,  and  S,  coincide,  the  heights  of  the  portions  of  the  section  of  curve 
e  e  have  to  be  [algebraically]  added  to  heights  of  the  section  of  curve  <i„8,  and 
similarly  for  the  sections  e  c,  and  8  S,.  The  result  of  this  addition  is  shown  in  the 
curve  C.  The  dotted  line  is  a  duplicate  of  the  section  d^S  in  the  curve  A.  Its 
object  is  to  make  the  composition  of  the  two  sections  immediately  evident  to  the 
eye.  It  is  easily  seen  that  the  curve  C  in  every  place  rises  as  much  above  or  sinks 
as  nnich  below  the  curve  A,  as  the  curve  B  respectively  rises  above  or  sinks 
beneath  the  horizontal  line.  The  heights  of  the  curve  C  are  consequently,  in  ac- 
cordance with  the  rule  for  compounding  vibrations,  equal  to  the  [algebraical]  sum 
of  the  corresponding  heights  of  A  and  B.  Thus  the  perpendicular  Ci  in  C  is  the  H 
sum  of  the  perpendiculars  a,  and  bi  in  A  and  B ;  the  lower  part  of  this  perpen- 
dicular Ci,  from  the  straight  line  up  to  the  dotted  curve,  is  equal  to  the  perpen- 
dicular ai,  and  the  upper  part,  from  the  dotted  to  the  continuous  curve,  is  equal  to 
the  perpendicular  bj.  On  the  other  hand,  the  height  of  the  perpendicular  Cj  is 
equal  to  the  height  a^  diminished  by  the  depth  of  the  fall  bo.  And  in  the  same 
way  all  other  points  in  the  curve  C  are  found.* 

It  is  evident  that  the  motion  represented  by  the  curve  C  is  also  periodic,  and 
that  its  periods  have  the  same  duration  as  those  of  A.  Thus  the  addition  of  the 
section  d„S  of  A  and  e  e  of  B,  must  give  the  same  result  as  the  addition  of  the 
perfectly  equal  and  similar  sections  8  8,  and  e  e„  and,  if  we  supposed  both  curves 
to  be  continued,  the  same  would  be  the  case  for  all  the  sections  into  which  they 
would  be  divided.  It  is  also  evident  that  equal  sections  of  both  curves  could  not 
continually  coincide  in  this  way  after  completing  the  addition,  unless  the  ciu'ves  thus 
added  could  be  also  separated  into  exactly  equal  and  similar  sections  of  the  same  II 
length,  as  is  the  case  in  fig.  11,  whei-e  two  periods  of  B  last  as  long  or  have  the 
same  horizontal  length  as  one  of  A.  Now  the  horizontal  lengths  of  our  figure 
represent  time,  and  if  we  pass  from  the  curves  to  the  real  motions,  it  results  that 
the  motion  of  air  caused  by  the  composition  of  the  two  simple  tones,  A  and  B,  is 
also  periodic,  just  because  one  of  these  simple  tones  makes  exactly  twice  as  many 
vibrations  as  the  other  in  the  same  time. 

It  is  easily  seen  by  this  example  that  the  peculiar  form  of  the  two  curves  A 
and  B  has  nothing  to  do  with  the  fact  that  their  sum  C  is  also  a  periodic  curve. 
Whatever  be  the  form  of  A  and  B,  provided  that  each  can  be  separated  into  equal 
and  similar  sections  which  have  the  same  horizontal  lengths  as  the  equal  and 
similar  sections  of  the  other — no  matter  whether  these  sections  correspond  to  one 
or  two,  or  three  periods  of  the  individual  curves — then  any  one  section  of  the  curve 
A  compounded  with  any  one  section  of  the  curve  B,  will  always  give  a  section 
of  the  curve  C,  which  will  have  the  same  length,  and  will  be  precisely  equal  and  •fl 
similar  to  any  other  section  of  the  curve  C  obtained  by  compounding  any  other 
section  of  A  with  any  other  section  of  B. 

When  such  a  section  embraces  several  periods  of  the  corresponding  curve  (as  in 
fig.  11,  the  sections  e  e  and  e  e,  each  consist  of  two  periods  of  the  simple  tone  B), 
then  the  pitch  of  this  second  tone  B,  is  that  of  an  upper  partial  tone  of  a  prime 
(as  the  simple  tone  A  in  fig.  11),  whose  period  has  the  length  of  that  principal 
section,  in  accordance  with  the  rule  above  cited. 

In  order  to  give  a  slight  conception  of  the  multiplicity  of  forms  producible  by 
comparatively  simple  compositions,  I  may  remark  that  the  compound  curve  would 

*  [Readers   not  used  to  geometrical  con-  spending  perpendiculars  in  A  and  B  in  proper 

structions  are  strongly  recommended  to  trace  directions,  and  joining  the  extremities  of  the 

the  two  curves  A  and  B,  and  to  construct  the  lengths  thus  found  by  a  curved  line.      In  this 

curve  C  from  them,  by  drawing  a  number  of  way  only  can  a  clear  conceiDtion  of  the  cona- 

perpendiculars  to  a  straight  line,  and   then  position  of  vibrations  be  rendcnnl  sufficiently 

setting  oS.  upon  them  the  lengths  of  the  corre-  familiar  for  subsequent  use. — Translatvr.'^ 


32 


DIFFERENCE  OF  PHASE. 


receive  another  form  if  the  curves  B,  fig.  11,  were  displaced  a  little  with  respect  to 
the  curve  A  before  the  addition  were  commenced-.  Let  B  be  displaced  by  being 
slid  to  the  right  until  the  point  e  falls  under  dj  in  A,  and  the  composition  will  then 
give  the  curve  D  with  narrow  crests  and  broad  troughs,  both  sides  of  the  crest 
being,  however,  equally  steep  ;  whereas  in  the  curve  C  one  side  is  steeper  than  the 
other.'  If  we  displace  the  curve  B  still  more  by  sliding  it  to  the  right  till  e  falls 
under  do,  the  compound  curve  would  resemble  the  reflection  of  C  in  a  mirror : 
that  is,  it  would  have  the  same  form  as  C  reversed  as  to  right  and  left ;  the  steeper 
inclination  which  in  C  lies  to  the  left  would  now  lie  to  the  right.  Again,  if  we 
displace  B  till  e  falls  under  dj  we  obtain  a  curve  similar  to  D,  fig.  11,  but  reversed 
as  to  up  and  down,  as  may  be  seen  by  holding  the  book  upside-down,  the  crests 
being  broad  and  the  troughs  narrow. 

Fig.  12. 


^ 


All  these  curves  with  their  various  transitional  forms  are  periodic  curves. 
Other  composite  periodic  curves  are  shown  at  C,  D,  fig.  12  above,  where  they  are 
compounded  of  the  two  curves  A  and  B,  having  their  periods  in  the  ratio  of  1  to  3. 
The  dotted  curves  are  as  before  copies  of  the  first  complete  vibration  or  period 
of  the  curve  A,  in  order  that  the  reader  may  see  at  a  glance  that  the  compound 
curve  is  always  as  much  Iiigher  or  lower  than  A,  as  B  is  higher  or  lower  than  the 
horizontal  line.  In  C,  the  curves  A  and  B  are  added  as  they  stand,  but  for  D  the 
curve  B  has  been  first  slid  half  a  wave's  length  to  the  right,  and  then  the  addition 
•j  has  been  effected.  Both  forms  differ  from  each  other  and  from  all  preceding  ones. 
C  has  broad  crests  and  broad  troughs,  D  narrow  crests  and  narrow  troughs. 

In  these  and  similar  cases  we  have  seen  that  the  compound  motion  is  perfectly 
and  regiilarly  periodic,  that  is,  it  is  exactly  of  the  same  kind  as  if  it  proceeded 
from  a  single  musical  tone.  The  curves  compounded  in  these  examples  correspond 
to  the  motions  of  single  simple  tones.  Thus,  the  motions  shown  in  fig.  11  (on 
p.  SOb,  c)  might  have  been  produced  by  two  tuning-forks,  of  which  one  soimded  an 
Octave  higher  than  the  other.  But  we  shall  hereafter  see  that  a  flute  by  itself 
when  gently  blown  is  sufticient  to  create  a  motion  of  the  air  corresponding  to  that 
shown  in  C  or  D  of  fig.  11.  The  motions  of  fig.  12  might  be  produced  by  two 
tuning-forks  of  which  one  sounded  the  twelfth  of  the  other.  Also  a  single  closed 
organ  pipe  of  the  narrower  kind  (the  stop  called  Quintnten*)  would  give  nearly  the 
same  motion  as  that  of  C  or  D  in  fig.  12. 


*  [The  names   of   the  stops    on    German 
organs  do  not  always  agree  with  those   on 


English  organs.     I  find  it  best,  therefore,  not 
to  translate  them,  but  to  give  their  explana- 


CHAP.  n.  ANALYSIS  INTO  SIMPLE  VIBIUTIONS.  33 

Here,  then,  the  motion  of  the  air  in  the  aural  passage  has  no  property  by  whicli 
tlie  composite*  musical  tone  can  be  distinguished  from  the  single  nuisical  tone. 
If  the  ear  is  not  assisted  by  other  accidental  circumstances,  as  by  one  tuning-fork 
beginning  to  sound  before  the  other,  so  that  we  hear  them  struck,  or,  in  the  other 
case,  the  rustling  of  the  wind  against  the  mouthpiece  of  the  flute  or  lip  of  the 
organ  pipe,  it  has  no  means  of  deciding  whether  the  musical  tone  is  sim])le  or 
composite. 

Now,  in  what  relation  does  the  ear  stand  to  such  a  motion  of  the  air  1  Does 
it  analyse  it  or  does  it  not  ?  F.xperience  shows  us  that  when  two  ti;ning-forks,  an 
Octave  or  a  Twelfth  apart  in  pitch,  are  sounded  together,  the  ear  is  quite  able  to 
distinguish  their  simple  tones,  although  the  distinction  is  a  little  more  difficult 
with  these  than  with  other  intervals.  But  if  the  ear  is  able  to  analyse  a  compo- 
site musical  tone  produced  by  two  tuning-forks,  it  cannot  but  be  in  a  condition  to 
carry  out  a  similar  analysis,  when  the  same  motion  of  the  air  is  produced  by  a  H 
single  flute  or  organ  pipe.  And  this  is  really  the  case.  The  single  musical  tone 
of  such  instruments,  proceeding  from  a  single  source,  is,  as  we  have  already  men- 
tioned, analysed  into  partial  simple  tones,  consisting  in  each  case  of  a  prime  tone, 
and  one  upper  partial  tone,  the  latter  being  different  in  the  two  cases. 

The  analysis  of  a  single  musical  tone  into  a  series  of  partial  tones  depends, 
then,  \ipon  the  same  property  of  the  car  as  that  which  enables  it  to  distinguish 
different  musical  tones  from  each  other,  and  it  miist  necessaril}'^  effect  both  analyses 
by  a  rule  which  is  independent  of  the  fact  that  the  waves  of  sound  are  produced 
by  one  or  by  several  musical  instruments. 

The  rule  by  which  the  ear  proceeds  in  its  anal^'sis  was  first  laid  down  as 
generally  true  by  G.  S.  Ohm.  Part  of  this  rule  has  been  already  enunciated  in 
the  last  chapter  (p.  2'3a),  where  it  was  stated  that  only  that  particular  motion  of 
the  air  which  we  have  denominated  a  simple  vibration,  for  whicli  the  vibrating 
particles  swing  backwards  and  forwards  according  to  the  law  of  pendular  motion,  H 
is  capable  of  exciting  in  the  ear  the  sensation  of  a  single  simple  tone.  Every 
motion  of  the  air,  then,  which  corresp07ids  to  a  compiosite  mass  of  nntsical  tones, 
is,  according  to  Ohtn^s  laio,  capable  of  being  analysed  into  a  sum  of  simple  pen- 
dular vibrations,  and  to  each  such  single  simple  vibration  corresponds  a  simjile 
tone,  sensible  to  the  ear,  and  having  a  pntch  detertnined,  by  the  periodic  time  of  the 
correspjonding  motion  of  the  air. 

The  proofs  of  the  correctness  of  this  law,  the  reasons  why,  of  all  vibrational 
forms,  only  that  one  which  we  have  called  a  simple  vibration  plays  such  an 
important  part,  must  be  left  for  Chapters  IV.  and  VI.  Our  present  business  is 
only  to  gain  a  clear  conception  of  what  the  rule  means. 

The  simple  vibrational  form  is  inalterable  and  always  the  same.  It  is  only  its 
amplitude  and  its  periodic  time  which  are  subject  to  change.  But  we  have  seen 
in  figs.  11  and  12  (p.  306  and  p.  326)  what  varied  forms  the  composition  of  only  two 
simple  vibrations  can  produce.  The  number  of  these  forms  might  be  greatly  in-  %. 
creased,  even  without  introducing  fresh  simple  vibrations  of  different  periodic 
times,  by  merely  changing  the   proportions  wdiich  the  heights  of  the  two  simple 

tions   from    E.    J.    Hopkins's    The   Organ,  its  iu  other  cases,  '  a  pipe  for  sounding  the  Twelfth 

History   and  Construction,    1870,    pp.  444-448.  in  addition  to  the  fundamental  tone'.  It  seems 

In   this   case   Mr.    Hopkins,  following   other  to   be   properly   the    English    stop    '  Tii:elfUi, 

authorities,  prints  the  word  '  quintato;;,'  and  Octave  Quin',  Ihioi/fciinu,'  No.  611,  p.  141  of 

defines  it,  in  IG  feet  tone,  as  '  double  stopped  Hopkins. —  Tnni^hitur.] 

diapason,  of  rather  small  scale,  producing  the  *  [The   reader   must   distinguish   between 

Twelfth  of  the  fundamental  sound,  as  well  as  single  and  simple  musical  tones.     A  single  tone 

the  ground-tone  itself,  that  is,  sounding  the  may  be  a  conqknuul  tone  inasmuch  as  it  may 

IG  and  5|  ft.  tones '  which  means  sounding  the  be  compounded  of  several  simple  musical  tones, 

notes  beginning  with  C'„  simultaneously  with  but  it  is  si)igle  because  it  is  produced  by  one 

tlie  notes  beginning  with  G,  which  is  called  the  sounding  body.     A  composite _  musical  tone  is 

5^  foot  tone,  because  according  to  the  organ-  necessarily  compound,  but  it  is  called  coinjiosite 

makers'  theory  ^not  practice)  the  length  of  the  because  it  is  made  up  of  tones  (simple  orcom- 

G  pipe  is  \  of  the  length  of  the  0  pipe,  and  ^  of  pound)  produced  by  several  sounding  bodies.— 

16  is  5J.      [See  p.  IM,  noteJ.J     And  similarly,  Translator.] 

D 


34  ANALYSIS  INTO  SIMPLE  VIBRATIONS.  part  i. 

vibrational  curves  A  and  B  bear  to  eacb  other,  or  displacing  the  curve  B  by  other 
distances  to  the  right  or  left,  than  those  ah-eadj  selected  in  the  figures.  By  these 
simplest  possible  examples  of  such  compositions,  the  reader  will  be  able  to  form 
some  idea  of  the  enormous  variety  of  forms  which  would  result  from  using  more 
than  two  simple  forms  of  vibration,  each  form  representing  an  upper  partial  tone 
of  the  same  prime,  and  hence,  on  addition,  always  producing  fresh  periodic  curves. 
We  should  be  able  to  make  the  heights  of  each  single  simple  vibrational  curve 
greater  or  smailer  at  pleasure,  and  displace  each  one  separately  by  any  amount  in 
respect  to  the  prime, — or,  in  physical  language,  we  should  be  able  to  alter  their 
amplitudes  and  the  difterence  of  their  phases ;  and  each  such  alteration  of  ampli- 
tude and  difference  of  phase  in  each  one  of  the  simple  vibrations  would  produce  a  fresli 
change  in  the  resulting  composite  vibrational  form.     [See  App.  XX.  sect.  M.  No.  2.1 

The  multiplicity  of  vibrational  forms  which  can  be  thus  produced  by  the  com- 
11  position  of  simple  pendular  vibrations  is  not  merely  extraordinarily  great :  it  is  so 
great  that  it  cannot  be  greater.  The  French  mathematician  Fourier  has  proved 
the  correctness  of  a  mathematical  law,  which  in  reference  to  our  present  subject 
may  be  thus  enunciated :  Any  given  regular  periodic  form  of  vibration  can 
always  be  produced  by  the  addition  of  simple  vibrations,  having  2^itch  numbers 
which  are  once,  twice,  thrice,  four  times,  dx.,  as  great  as  the  pitch  nwnbers  of  the 
given  motion. 

The  amplit%ides  of  the  elementary  simple  vibrations  to  which  the  height  of  our 
wave-curves  corresponds,  and  the  difference  of  phase,  that  is,  the  relative  amount 
of  horizontal  displacement  of  the  wave-curves,  can  always  be  found  in  every  given 
case,  as  Fourier  has  shown,  by  peculiar  methods  of  calculation  (which,  however, 
do  not  admit  of  any  popular  explanation),  so  that  any  given  regularly  periodic 
motion  can  always  be  exhibited  in  one  single  way,  and  in  no  other  way  whatever, 
as  the  sum  of  a  certain  number  of  j^endidar  vibrations. 
tI  Since,  according  to  the  results  already  obtained,  any  regularly  periodic  motion 
corresponds  to  some  musical  tone,  and  any  simple  pendular  vibration  to  a  simple 
musical  tone,  these  propositions  of  Fourier  ma\  be  thus  expressed  in  acoustical 
terms : 

Any  vibrational  motion  of  the  air  in  the  entrance  to  the  ear,  corresjwnding  to  a 
musical  tone,  may  be  cdtvays,  and  for  each  case  only  in  one  single  way,  exhibited  as 
the  sum  of  a  number  of  simjde  vibrational  motions,  corresponding  to  the  partials 
of  this  musical  tone. 

Since,  according  to  these  propositions,  any  form  of  vibration,  no  matter  what 
shape  it  maj^  take,  can  be  expressed  as  the  sum  of  simple  vibrations,  its  analysis 
into  such  a  sum  is  quite  independent  of  the  power  of  the  eye  to  perceive,  by  looking 
at  its  representative  curve,  whether  it  contains  simple  vibrations  or  not,  and  if  it 
does,  what  they  are.  I  am  obliged  to  lay  stress  upon  this  point,  because  I  have  by 
no  means  unfrequently  found  even  physicists  start  on  the  false  hypothesis,  that  the 
H  vibrational  form  must  exhibit  little  waves  corresponding  to  the  several  audible 
upper  partial  tones.  A  mere  inspection  of  the  figs.  II  and  12  (p.  30b  and  p.  32b) 
will  suffice  to  show  that  although  the  composition  can  be  easily  traced  in  the  parts 
where  the  curve  of  the  prime  tone  is  dotted  in,  this  is  quite  impossible  in  those 
parts  of  the  curves  C  and  D  in  each  figure,  where  no  such  assistance  has  been 
provided.  Or,  if  we  suppose  that  an  observer  who  had  rendered  himself  thoroughly 
familiar  with  the  curves  of  simple  vibrations  imagined  that  he  could  trace  the  com- 
position in  these  easy  cases,  he  would  certainly  utterly  fail  on  attempting  to  dis- 
cover by  his  eye  alone  the  composition  of  such  curves  as  are  shown  in  figs.  8 
and  9  (p.  21c).  In  these  will  be  found  straight  lines  and  acute  angles.  Perha]js 
it  will  be  asked  how  it  is  possible  by  compounding  such  smooth  and  uniformly 
rounded  curves  as  those  of  our  simple  vibrational  forms  A  and  B  in  figs.  1 1  and 
12,  to  generate  at  one  time  straight  lines,  and  at  another  acute  angles.  The 
answer  is,  that  an  infinite  number  of  simple  vibrations  are  required  to  generate 
curves  with   such  discontinuities  as  are  there  shown.      But  when  a  great  many 


CHAP.  II.  ANALYSIS  INTO  SIMPLE  VIBRATIONS.  3o 

such  curves  are  combined,  and  are  so  chosen  that  in  certain  places  thev  all  bend 
in  the  same  direction,  and  in  oth^rs'in  opposite  directions,  the  curvatures  mutually 
strengthen  each  other  in  the  first  case,  finally  producing  an  infinitely  great  curva- 
ture, that  is,  an  acute  angle,  and  in  the  second  case  they  mutually  weaken  each 
other,  so  that  ultimately  a  straight  line  results.  Hence  we  can  generally  lay  it 
down  as  a  rule  that  the  force  or  loudness  of  the  upper  partial  tones  is  the  gi'eater, 
the  sharper  the  discontinuities  of  the  atmospheric  motion.  When  the  motion 
alters  uniformly  and  gradually,  answering  to  a  vibrational  curve  j)roceeding  in 
smoothly  curved  forms,  only  the  deeper  partial  tones,  which  lie  nearest  to  the 
prime  tone,  have  any  perceptible  intensity.  But  where  the  motion  alters  by  jumps, 
and  hence  the  vibrational  curves  show  angles  or  sudden  changes  of  curvature,  the 
upper  partial  tones  will  also  have  sensible  force,  although  in  all  these  cases  the 
amplitudes  decrease  as  the  pitch  of  the  upper  partial  tones  becomes  higher.* 

We  shall  become  acquainted  with  examples  of  the  analysis  of  given  vibrational  H 
forms  into  separate  partial  tones  in  Chapter  V. 

The  theorem  of  F'ourier  here  adduced  shows  first  that  it  is  mathematicallv 
possible  to  consider  a  musical  tone  as  a  sum  of  simple  tones,  in  the  meaning  we 
have  attached  to  the  words,  and  mathematicians  have  indeed  always  found  it 
convenient  to  base  their  acoustic  investigations  on  this  mode  of  analysing  vibrations. 
But  it  by  no  means  follows  that  we  are  obliged  to  consider  the  matter  in  this  way. 
We  have  rather  to  inquire,  do  these  partial  constituents  of  a  musical  tone,  such  as 
the  mathemathical  theory  distinguishes  and  the  ear  perceives,  really  exist  in  the 
mass  of  air  external  to  the  ear?  Is  this  means  of  analysing  forms  of  vibration 
which  Fourier's  theorem  prescribes  and  renders  possible,  not  merely  a  mathematical 
fiction,  permissible  for  facilitating  calculation,  but  not  necessarily  having  any 
corresponding  actual  meaning  in  things  themselves?  What  makes  us  hit  upon 
pendidar  vibrations,  and  none  other,  as  the  simplest  element  of  all  motions  pro- 
ducing sound  ?  We  can  conceive  a  whole  to  be  split  into  parts  in  very  different 
and  arbitrary  ways.  Thus  we  may  find  it  convenient  for  a  certain  calcxdation  to  ^ 
consider  the  number  12  as  the  sum  8  +  4,  because  the  8  may  have  to  be  cancelled, 
but  it  does  not  follow  that  12  must  always  and  necessarily  be  considered  as  merely 
the  sum  of  8  and  4.  In  another  case  it  might  be  more  convenient  to  consider  12 
as  the  sum  of  7  and  5.  Just  as  little  does  the  mathematical  possibility,  proved  by 
Fourier,  of  compoimding  all  pei'iodic  vibrations  out  of  simple  vibrations,  justifv 
us  in  concluding  that  this  is  the  only  permissible  form  of  analysis,  if  we  cannot  in 
addition  establish  that  this  analysis  has  also  an  essential  meaning  in  nature.  That 
this  is  indeed  the  case,  that  this  analysis  has  a  meaning  in  nature  independently 
of  theory,  is  rendered  probable  by  the  fact  that  the  ear  really  effects  the  same 
analysis,  and  also  by  the  circumstance  already  named,  that  this  kind  of  analysis 
has  been  found  so  much  more  advantageous  in  mathematical  investigations  than 
any  other.  Those  modes  of  regarding  phenomena  that  correspond  to  the  most 
intimate  constitution  of  the  matter  under  investigation  are,  of  course,  also  always 
those  which  lead  to  the  most  suitable  and  evident  theoretical  treatment.  But  it  ^ 
would  not  be  advisable  to  begin  the  investigation  with  the  functions  of  the  ear, 
because  these  are  very  intricate,  and  in  themselves  require  much  explanation. 
In  the  next  chapter,  therefore,  we  shall  inquire  whether  the  analysis  of  compound 
into  simple  vibrations  has  an  actually  sensible  meaning  in  the  external  world, 
independently  of  the  action  of  the  ear,  and  we  shall  really  be  in  a  condition  to 
show  that  certain  mechanical  effects  depend  upon  whether  a  certain  partial  tone 

*  Supposing   n    to   be  the  number  of  the       a  sudden  jump, .and  hence  the  curve  lias  an 
order  of  a  partial  tone,  and  n  to  be  very  large,  1 

then  the  amplitude  of  the  upper  partial  tones       acute  angle  ;  (3)  as  ^-^-^,  when  the  curvature 

decreases :  (1)  as  -,  when  tbe  amplitude  of  the  f^^f'  suddenly  ;  (4)  when  none  of  the  differen- 

w  '■  tial  quotients  are   discontniuous,   they  must 

vibrations  themselves  makes  a  sudden  jump ;  ,  j.  i       j.       ^    i.         -«- 

"1^  J       i  '  decrease  at  least  as  fast  as  c     . 

(21as  — ,  when  their  differential  quotient  makes 


36  MECHANICS  OF  SYMPATHETIC  RESONANCE.  paut  i. 

is  or  is  not  contained  in  a  composite  mass  of  musical  tones.  Tlie  existence 
of  partial  tones  will  thus  acquire  a  meaning  in  nature,  and  our  knowledge  of 
their  mechanical  effects  will  in  turn  shed  a  new  light  on  their  relations  to  the 
human  ear. 


CHAITEH   III. 

ANALYSIS    OF    MUSICAL    TONES    CY    SYMrATHETIC    RESONANCE. 

AVE  proceed  to  show  that  the  simple  partial  tones  contained  in  a  composite  mass 
of  musical  tones,  produce  peculiar  mechanical  effects  in  nature,  altogether  inde- 
pendent of  the  human  ear  and  its  sensations,  and  also  altogether  independent  of 

^  merely  theoretical  considerations.  These  effects  consequently  give  a  peculiar  objec- 
tive significance  to  this  peculiar  method  of  analysing  vibrational  forms. 

Such  an  effect  occurs  in  the  phenomenon  of  sympathetic  resonance.  This 
phenomenon  is  always  found  in  those  bodies  which  when  once  set  in  motion  by 
any  impiilse,  continue  to  perform  a  long  series  of  vibrtitions  before  they  come  to 
rest.  When  these  bodies  are  struck  gently,  but  periodically,  although  each  blow 
may  be  separately  quite  insufficient  to  produce  a  sensible  motion  in  the  vibratory 
body,  yet,  provided  the  periodic  time  of  the  gentle  blows  is  precisely  the  same  as 
the  periodic  time  of  the  body's  own  vibrations,  very  large  and  powerful  oscilla- 
tions may  result.  But  if  the  periodic  time  of  the  regular  blows  is  different  from 
the  periodic  time  of  the  oscillations,  the  resulting  motion  will  be  weak  or  quite 
insensible. 

Periodic  impulses  of  this  kind  generally  proceed  from  another  body  which  is 
already  vibrating  regularly,  and  in  this  case  the  swings  of  the  latter  in  the  course 

H  of  a  little  time,  call  into  action  the  swings  of  the  former.  Under  these  circum- 
stances we  have  the  pi'ocess  called  syrnjmthetic  oscillation  or  sympathetic  resonance. 
The  essence  of  the  mechanical  effect  is  independent  of  the  rate  of  motion,  which 
may  be  fast  enough  to  excite  the  sensation  of  sound,  or  slow  enough  not  to  produce 
anything  of  the  kind.  Musicians  are  well  acquainted  with  symjmthetic  resonance. 
When,  for  example,  the  strings  of  two  violins  are  in  exact  unison,  and  one  string  is 
bowed,  the  other  will  begin  to  vibrate.  But  the  nature  of  the  process  is  best  seen 
in  instances  where  the  vibrations  are  slow  enough  for  the  eye  to  follow  the  whole 
of  their  successive  phases. 

Thus,  for  example,  it  is  known  that  the  largest  clnu-ch-bells  may  be  set  in  motion 
by  a  man,  or  even  a  boy,  who  pulls  the  ropes  attached  to  them  at  proper  aiid  regular 
intervals,  even  when  their  weight  of  metal  is  so  great  that  the  strongest  man  could 
scarcely  move  them  sensibly,  if  he  did  not  apply  his  strength  in  determinate 
periodical  intervals.     When  such  a  l»ell  is  once  set  in  motion,  it  continues,  like  a 

H  struck  pendulum,  to  oscillate  for  some  time,  until  it  gradually  returns  to  rest,  even 
if  it  is  left  quite  by  itself,  and  no  force  is  employed  to  arrest  its  motion.  The 
motion  diminishes  gradually,  as  we  know,  because  the  friction  on  the  axis  and  the 
resistance  of  the  air  at  every  swing  destroy  a  portion  of  the  existing  moving  force. 

As  the  bell  swings  backwards  and  forwards,  the  lever  and  vo\)e  fixed  to  its  axis 
rise  and  fall.  If  when  the  lever  falls  a  boy  clings  to  the  lower  end  of  the  bell-rope, 
his  weight  will  act  so  as  to  increase  the  rapidity  of  the  existing  motion.  This 
increase  of  velocity  may  be  very  small,  and  yet  it  will  produce  a  coi-responding 
increase  in  the  extent  of  the  bell's  swings,  which  again  will  continue  for  a  while, 
until  destroyed  by  the  friction  and  resistance  of  the  air.  But  if  the  boy  clung  to  the 
bell-rope  at  a  wrong  time,  while  it  Avas  ascending,  for  instance,  the  weight  of  his 
body  would  act  in  opposition  to  the  motion  of  the  bell,  and  the  extent  of  swing 
Avould  decrease.  Now,  if  the  boy  continued  to  cling  to  the  rope  at  each  swing  so 
long  as  it  was  falling,  and  then  let  it  ascend  freely,  at  every  swing  the  motion  of 
the  bell  would  be  only  increased  in  speed,  and  its  swings  would  gradually  become 


CHAP.  III.  MKCHANICS  OF  SYMPATHETIC  RESONANCE.  37 

greater  and  greater,  \iiitil  by  their  increase  the  motion  imparted  on  every  osciUation 
of  the  bell  to  the  walls  of  the  belfry,  and  the  external  air  would  become  so  great 
as  exactly  to  be  covered  by  the  power  exerted  by  the  boy  at  each  swing. 

The  success  of  this  process  depends,  therefore,  essentially  on  the  boy's  applying 
his  force  only  at  those  moments  when  it  will  increase  the  motion  of  the  bell.  That 
is,  he  must  employ  his  strength  periodically,  and  the  ])criodic  time  must  be  equal 
to  that  of  the  bell's  swing,  or  he  will  not  be  successful.  He  would  just  as  easily 
bring  the  swinging  bell  to  rest,  if  he  clung  to  the  rope  only  during  its  ascent,  and 
thus  let  his  weight  be  raised  by  the  bell. 

A  similar  experiment  which  can  be  tried  at  any  instant  is  the  following.  Con- 
struct a  pendulum  by  hanging  a  heavy  body  (such  as  a  ring)  to  the  lower  end  of  a 
thread,  holding  the  upper  end  in  the  hand.  On  setting  the  ring  into  gentle  pen- 
dular  vibration,  it  will  be  found  that  this  motion  can  be  gradually  and  considerably 
increased  by  watching  the  moment  when  the  pendulum  has  reached  its  greatest  H 
departure  from  the  vertical,  and  then  giving  the  hand  a  very  small  motion  in  the 
opposite  direction.  Thus,  when  the  pendulum  is  furthest  to  the  right,  move  the 
hand  very  slightly  to  the  left ;  and  when  the  pendulum  is  furthest  to  the  left,  move 
the  hand  to  the  right.  The  pendulum  may  be  also  set  in  motion  from  a  state  of 
rest  by  giving  the  h^md  similar  very  slight  motions  having  the  same  periodic  time 
as  the  pendulum's  own  swings.  The  displacements  of  the  hand  may  be  so  small 
under  these  circumstances,  that  they  can  scarcely  be  perceived  with  the  closest 
attention,  a  circumstance  to  which  is  due  the  superstitious  application  of  this 
little  apparatus  as  a  divining  rod.  If  namely  the  observer,  without  thinking  of 
his  hand,  follows  the  swings  of  the  pendulum  with  his  eye,  the  hand  readily  follows 
the  eye,  and  involuntarily  moves  a  little  backwards  or  forwards,  precisely  in  the 
same  time  as  the  pendulum,  after  this  has  accidentally  begun  to  move.  These 
involuntary  motions  of  the  hand  are  usnally  overlooked,  at  least  when  the  observer 
is  not  accustomed  to  exact  observations  on  such  unobtrusive  influences.  By  this  "^ 
nieans  any  existing  vibration  of  the  pendulum  is  increased  and  kept  up,  and  any 
accidental  motion  of  the  ring  is  readily  converted  into  pendular  vibrations, 
which  seem  to  arise  spontaneously  without  any  co-operation  of  the  observer, 
and  are  hence  attributed  to  the  influence  of  hidden  metals,  running  streams,  and 
so  on. 

If  on  the  other  hand  the  motion  of  the  hand  is  intentionally  made  in  the  con- 
trary direction,  the  pendiilum  soon  comes  to  rest. 

The  explanation  of  the  process  is  very  simple.  When  the  iipper  end  of  the 
thread  is  fastened  to  an  immovable  support,  the  pendulum,  once  struck,  continues 
to  swhig  for  a  long  time,  and  the  extent  of  its  swings  diminishes  very  slowly.  We 
can  suppose  the  extent  of  the  swings  to  be  measured  by  the  angle  which  the  thread 
makes  with  the  vertical  on  its  greatest  deflection  from  it.  If  the  attached  body 
at  the  point  of  greatest  deflection  lies  to  the  right,  and  we  move  the  hand  to  the 
left,  we  manifestly  increase  the  angle  between  the  string  and  the  vertical,  and  con- 1 
sequently  also  augment  the  extent  of  the  swing.  By  moving  the  upper  end  of  the 
string  in  the  opposite  direction  we  should  decrease  the  extent  of  the  swmg. 

In  this  case  there  is  no  necessity  for  moving  the  hand  in  the  same  periodic  time 
as  the  pendulum  swings.  We  miglit  move  the  hand  backwards  and  forwards  only 
at  every  third  or  fifth  or  other  swing  of  the  pendulum,  and  we  should  still  produce 
large  swings.  Thus,  when  the  pendulum  is  to  the  right,  move  the  hand  to  the 
left,  and  keep  it  still,  till  the  pendulum  has  swung  to  the  left,  then  again  to  the 
right,  and  then  once  more  to  the  left,  and  then  return  the  hand  to  its  first  position, 
afterwards  wait  till  the  pendidum  has  swung  to  the  right,  then  to  the  left,  and 
again  to  the  right,  and  then  recommence  the  first  motion  of  the  hand.  In  this 
way  three  complete  vibrations,  or  double  excursions  of  the  pendulum,  will  corre- 
spond to  one  left  and  right  motion  of  the  hand.  In  the  same  way  one  left  and 
right  motion  of  the  hand  may  be  made  to  correspond  with  seven  or  more  swings 
of  the  pendulum.     The  meaning  of  this  i)rocess  is  always  that  the  motion  of  the 


38  MECHANICS  OF  SYMPATHETIC  RESONANCE.  part  i. 

liand  must  in  each  case  be  made  at  such  a  time  and  in  such  a  direction  as  to  be 
opposed  to  the  deflection  of  the  penduhun  and  consequently  to  inci'ease  it. 

By  a  sHght  alteration  of  the  process  we  can  easily  make  two,  four,  si.x,  etc., 
swings  of  the  pendulum  correspond  to  one  left  and  right  motion  of  the  hand  ;  for 
a  sudden  motion  of  the  hand  at  the  instant  of  the  pendulum's  passage  through  the 
vertical  has  no  influence  on  the  size  of  the  swings.  Hence  when  the  pendulum 
lies  to  the  right  move  the  hand  to  the  left,  and  so  increase  its  velocity,  let  it  swing 
to  the  left,  watch  for  the  moment  of  its  passing  the  vertical  line,  and  at  that  instant 
return  the  hand  to  its  original  position,  allow  it  to  reach  the  right,  and  then  again 
the  left  and  once  more  the  right  extremity  of  its  arc,  and  then  recommence  the 
first  motion  of  the  hand. 

We  are  able  then  to  communicate  violent  motion  to  the  pendulum  by  very 
small  periodical  vibrations  of  the  hand,  having  their  periodic  time  exactly  as  great, 
Hor  else  two,  three,  four,  &c.,  times  as  great  as  that  of  the  peudular  oscillation.  We 
have  here  considered  that  the  motion  of  the  hand  is  backwards.  This  is  not 
necessary.  It  may  take  place  continuously  in  any  other  way  we  please.  When  it 
moves  continuously  there  Avill  be  generally  portions  of  time  during  which  it  will 
increase  the  pendulum's  motion,  and  others  perhaps  in  which  it  will  diminish  the 
same.  In  order  to  create  strong  vibrations  in  the  pendulum,  then,  it  will  be 
necessary  that  the  increments  of  motion  should  l)e  permanently  predominant,  and 
should  not  be  neutralised  by  the  sum  of  the  decrements. 

Now  if  a  determinate  periodic  motion  were  assigned  to  the  hand,  and  we  wished 
to  discover  wliether  it  would  produce  considerable  vibrations  in  the  pendulum,  we 
could  not  alwaj's  predict  the  result  without  calculation.  Theoretical  mechanics 
would,  however,  prescribe  the  following  })i-ocess  to  be  pursued :  Analyse  the  lieriocUc 
motion  of  the  hand  into  a  sum  of  simple  pendular  vihrations  of  the  Aa?ifZ— exactly 
in  the  same  way  as  was  laid  down  in  the  last  chapter  for  the  periodic  motions  of 
51  the  particles  of  air, — then,  if  the  periodk  time  of  one  of  these  vibrations  is  eqwil 
to  the  periodic  time  of  the  2'Ctiduliim's  oi"n  oscillations,  the  j/enduliim  roill  be  set 
into  violent  motion,  but  not  otherwise.  We  might  compound  small  pendular 
motions  of  the  hand  out  of  viljrations  of  other  periodic  times,  as  much  as  we  liked, 
but  we  should  fail  to  produce  any  lasting  strong  swings  of  the  pendulum.  Hence 
the  analysis  of  the  motion  of  the  hand  into  pendular  swings  has  a  real  meaning  in 
nature,  producing  determinate  mechanical  effects,  and  for  the  present  purpose  no 
other  analysis  of  the  motion  of  the  hand  into  any  other  partial  motions  can  be 
substituted  for  it. 

In  the  above  examples  tlie  pendulum  could  lie  set  into  sympathetic  vibration, 
when  the  hand  moved  periodically  at  the  same  rate  as  the  pendulum ;  in  this  case 
the  longest  partial  vibration  of  the  hand,  corresponding  to  the  prime  tone  of  a 
resonant  vibration,  was,  so  to  sjjeak,  in  unison  with  the  pendulum.  When  three 
swings  of  the  pendulum  went  to  one  backwards  and  forwards  motion  of  the  hand, 
H  it  was  the  third  partial  swing  of  the  hand,  answering  as  it  were  to  the  Twelfth  of 
its  prime  tone,  which  set  the  pendulum  in  motion.     And  so  on. 

The  same  process  that  we  have  thus  become  acquainted  with  for  swings  of  long 
periodic  time,  holds  precisely  for  swings  of  so  short  a  period  as  sonorous  vibrations. 
Any  elastic  body  which  is  so  fastened  as  to  admit  of  continuing  its  vibrations  for 
some  length  of  time  when  once  set  in  motion,  can  also  be  made  to  vibrate  sym- 
p.itheticall}',  when  it  receives  periodic  agitations  of  comparatively  small  amounts, 
having  a  periodic  time  corresponding  to  that  of  its  own  tone. 

Gently  touch  one  of  the  keys  of  a  pianoforte  without  striking  the  string,  so  as 
to  raise  the  damper  only,  and  then  sing  a  note  of  the  corresponding  pitch  forcibly 
directing  the  voice  against  the  strings  of  the  instrument.  On  ceasing  to  sing,  the 
note  will  be  echoed  back  from  the  piano.  It  is  easy  to  discover  that  this  echo  is 
caused  by  the  string  which  is  in  unison  with  the  note,  for  directly  the  hand  is 
removed  from  the  key,  and  the  damper  is  allowed  to  fall,  tlie  echo  ceases.  The 
sympathetic  vibration  of  the   sti-ing  is  still  better  shown  by  putting  little  paper 


CHAP.  III.    DIFFERENT  EXTENT  OF  SYMPATHETIC  RESONANCE.  39 

riders  iipon  it,  which  are  jerked  oft"  as  soon  as  the  string  vibrates.  The  more 
exactly  the  singer  hits  the  pitch  of  the  string,  the  more  strongly  it  vibrates.  A 
very  little  deviation  from  the  exact  pitch  fails  in  exciting  sympathetic  vibration. 

In  this  experiment  the  sounding  board  of  the  instrument  is  first  struck  by  the 
vibrations  of  the  air  excited  by  the  human  voice.  The  sounding  board  is  well 
known  to  consist  of  a  broad  flexible  wooden  plate,  which,  owing  to  its  exten- 
sive siirface,  is  better  adapted  to  convey  the  agitation  of  the  strings  to  the  air, 
and  of  the  air  to  the  strings,  than  the  small  surface  over  which  string  and  air  arc 
themselves  directly  in  contact.  The  sounding  board  first  commiuiicates  the  agita- 
tions which  it  receives  from  the  air  excited  by  the  singer,  to  the  points  where  the 
string  is  fastened.  The  magnitude  of  an}-  single  such  agitation  is  of  course  infini- 
tesimally  small.  A  very  large  number  of  such  effects  must  necessarily  be  aggre- 
gated, before  any  sensible  motion  of  the  string  can  be  caused.  And  such  a  con- 
tinuous addition  of  eff"ects  really  takes  place,  if,  as  in  the  preceding  experiments  with  ^ 
the  bell  and  the  pendulum,  the  periodic  time  of  the  small  agitations  which  are  com- 
municated to  the  extremities  of  the  string  by  the  air,  through  the  intervention  of  the 
sounding  board,  exactl}-  corresponds  to  the  periodic  time  of  the  string's  own  vibra- 
tions. When  this  is  the  case,  a  long  series  of  such  vibrations  will  really  set  the 
string  into  motion  which  is  very  violent  in  comparison  with  the  exciting  cause. 

In  place  of  the  human  voice  we  might  of  course  use  any  other  musical  instru- 
ment. Provided  only  that  it  can  produce  the  tone  of  the  pianoforte  string  accu- 
rately and  sustain  it  powerfully,  it  will  bring  the  latter  into  sympathetic  vibration. 
In  place  of  a  pianoforte,  again,  we  can  employ  any  other  stringed  instrument 
having  a  sounding  board,  as  a  violin,  guitar,  harp,  etc.,  and  also  stretched  mem- 
branes, bells,  elastic  tongues  or  plates,  ifec,  provided  only  that  the  latter  are  so 
fastened  as  to  admit  of  their  giving  a  tone  of  sensible  duration  when  once  made 
to  sound.  ^ 

Wlien  the  pitch  of  the  original  sounding  body  is  not  exactly  that  of  the  sym- 
|)athising  body,  or  that  which  is  meant  to  vibrate  in  sympathy  with  it,  the  latter 
will  nevertheless  often  make  sensible  sympathetic  vibrations,  which  will  diminish 
in  amplitude  as  the  difference  of  pitch  increases.  But  in  this  respect  difterent 
sounding  bodies  show  great  difterences,  according  to  the  length  of  time  for  which 
they  continue  to  soiuid  after  having  been  set  in  action  before  comnnuiicating  their 
whole  motion  to  the  air. 

Bodies  of  small  mass,  which  readily  communicate  their  motion  to  the  air,  and 
quickly  cease  to  sound,  as,  for  example,  stretched  membranes,  or  violin  strings,  are 
readily  set  in  sympathetic  vibration,  because  the  motion  of  the  air  is  conversely 
readily  transferred  to  them,  and  they  are  also  sensibly  moved  by  sufficiently  strong 
agitations  of  the  air,  even  when  the  latter  have  not  precisely  the  same  periodic 
time  as  the  natural  tone  of  the  sympathising  bodies.  The  limits  of  pitch  capable 
of  exciting  sympathetic  vibration  are  consequently  a  little  wider  in  this  case.  By 
the  comparatively  greater  influence  of  the  motion  of  the  air  upon  light  elastic  H 
bodies  of  this  kind  which  offer  but  little  resistance,  their  natural  periodic  time  can 
be  slightly  altered,  and  adapted  to  that  of  the  exciting  tone.  Massive  elastic 
bodies,  on  the  other  hand,  which  are  not  readily  movable,  and  are  slow  in  com- 
municating their  sonorous  vibrations  to  the  air,  such  as  bells  and  ])lates,  and  con- 
tinue to  sound  for  a  long  time,  are  also  more  difficult  to  move  by  the  air.  A  much 
longer  addition  of  effects  is  required  for  this  purpose,  and  consequently  it  is  also 
necessary  to  hit  the  pitch  of  their  own  tone  with  much  greater  nicety,  in  order  to 
make  them  vibrate  sympathetically.  Still  it  is  well  known  that  bell-shaped  glasses 
can  be  put  into  violent  motion  by  singing  their  proper  tone  into  them  ;  indeed  it  is 
i-elated  that  singers  with  very  powerful  and  pure  voices,  have  sometimes  been  able 
to  crack  them  by  the  agitation  thus  caused.  The  principal  difficulty  in  this  experi- 
ment is  in  hitting  the  pitch  with  sufficient  precision,  and  retaining  the  tone  at  that 
exact  pitch  for  a  sufficient  length  of  time. 

Tuning-forks  are  the  most  difficult  bodies  to  set  in  sympathetic  \ibration.     To 


40     INFLUENCE  OF  PAUTIALS  ON  SYMl'ATHETIC  RESONANCE,     part  i. 

effect  this  they  may  be  fastened  on  sounding  boxes  which  have  been  exactly  tuned  to 

their  tone,  as  shown  in  fig.  13.     If  we  have  two  such  forks  of  exactly  the  same 

pitch,  and  excite  one  by  a  violin  bow, 

the  other  will  begin  to  vibrate  in  sym- 
pathy,  even   if  placed   at   the  further 

end  of  the  same  room,  and  it  will  con- 
tinue to  sound,  after  the  first  has  been 

damped.       The  astonishing  nature  of 

such  a  case  of  sympathetic  vibration 

will  appear,  if  we  merely  compare  the 

heavy  and  powerful  mass  of  steel  set 

in  motion,  with  the  light  yielding  mass 

of  air  which  produces  the  effect  by  such 
U  small  motive  powers  that  they  could 

not  stir  the  lightest  spring  which  was 

not  in  tune  with  the  fork.     With  such 

forks  the   time   required   to   set   them 

in   full   swing   by  sympathetic   action, 

is  also  of  sensible   duration,   and   the 

slightest  disagreement  in  pitch  is  sufficient   to  produce  a  sensible  diminution   in 

the  sympathetic  effect.     By  sticking  a  piece  of  wax  to  one  prong  of  the  second 

fork,  sufficient  to  make  it  vibrate  once  in  a  second  less  than  the  first — a  difference 

of   pitch    scarcely  sensible    to   the    finest   car — the   sympathetic  vibration   will   be 

wholly  destroyed. 

After    having    thus   described    the    phenon\enon    of    sympathetic    vibration    in 

general,  we  proceed  to  investigate  the  influence  exerted   in  sympathetic  resonance 

l)y  the  different  forms  of  wave  of  a  musical  tone. 
^        First,  it  must  be  observed  that  most  elastic  bodies  which  have  been  set  into 

sustained  vibration  by  a  gentle  force  acting  periodically,  are  (with  a  few  exceptions 


to  be  considered  hereafter)  always  made  to  swing  in  pendular  vibrations.  But  they 
are  in  general  capable  of  executing  several  kinds  of  such  vibration  with  different 
periodic  times  and  with  a  different  distribution  over  the  various  i)arts  of  the 
vibrating  body.  Hence  to  the  different  lengths  of  the  periodic  times  correspond 
different  simple  tones  producible  on  such  an  elastic  body.  These  are  its  so-called 
proper  tones.  It  is,  however,  only  exceptionally,  as  in  strings  and  the  narrower 
kinds  of  organ  pipes,  that  these  proper  tones  correspond  in  pitch  with  the  har- 


III.    INFLUENCE  OF  PARTIALS  ON  SYMPATHETIC  RESONANCE.      41 


uionic  upper  i)artial  tones  of  a  musical  tone  already  mentioned.     They  are  for  tl>e 
most  part  inharmonic  in  relation  to  the  prime  tone. 

In  many  cases  the  vibrations  and  their  mode  of  distribution  over  the  vibrating 
bodies  can  be  rendered  visible  by  strewing  a  little  fine  sand  over  the  latter.  Take,  for 
■example,  a  menibrane  (as  a  bladder  or  piece  of  thin  india-rubber)  stretched  over  a 
circular  ring.  In  fig.  14  are  shown  the  various  forms  which  a  membrane  can 
jissume  when  it  vibrates.  The  diameters  and  circles  on  the  surface  of  the  mem- 
brane mark  those  points  which  remain  at  rest  during  the  vibration,  and  are  known 
<is  nodal  linen.  By  these  the  surface  is  divided  into  a  number  of  compartments 
which  bend  altei-nately  up  and  down,  in  such  a  way  that  while  those  marked  ( + ) 
rise,  those  marked  (-)  fall.  Over  the  figures  a,  b,  c,  are  shown  the  forms  of  a 
.section  of  the  membrane  during  vibration.  Only  those  forms  of  motion  are  drawn 
which  correspond  with  the  deepest  and  most  easily  producible  tones  of  the  mem- 
l)rane.  The  number  of  circles  and  diameters  can  be  increased  at  pleasure  by  51 
taking  a  sufficiently  thin  membrane,  and  stretching  it  with  sufiicient  regularity, 
and  in  this  case  the  tones  would  continually  sharpen  in  pitch.  By  strewing  sand 
on  the  membrane  the  figures  are  easily  rendered  visible,  for  as  soon  as  it  begins 
to  vibrate  the  particles  of  sand'collect  on  the  nodal  lines. 

In  the  same  way  it  is  possible  to  render  visible  the  nodal  lines  and  forms  of 
vibration  of  oval  and  square  membranes,  and  of  differently-shaped  plane  elastic" 
plates,  bars,  and  so  on.  These  form  a  series  of  very  interesting  phenomena  dis- 
covered by  Chladni,  but  to  pursue  them  would  lead  us  too  far  from  our  proper 
.subject.  It  will  suflice  to  give  a  few  details  respecting  the  simplest  case,  that  of  a 
circular  memlirane. 

In  the  time  required  by  the  membrane  to  execute   100  vibrations  of  the  form  a, 
fig.     14    (p.    40c),   the   number  of  vibrations   executed   by  the    other   forms   is    as 
follows  : — 
H 


Form  of  \'ibration 


a  without  uodal  lines    . 

b  with  one  circle  .... 

c  with  two  circles 

d  with  one  diameter 

e  with  one  diameter  and  one  circle 

f  with  two  diameters    . 


Pitch  Number 

Cents  * 

Notes  nearly 

100 

0 

(. 

229-6 

1439 

rf'-f- 

859-9 

2217 

h'h  + 

159 

805 

«[? 

292 

1858 

'j\r 

214 

1317 

4+ 

The  prime  tone  has  been  here  arbitrarily  assumed  as  c,  in  order  to  note  the  inter- 
vals of  the  higher  tones.  Those  simple  tones  produced  by  the  membrane  which  are 
slightly  higher  than  those  of  the  note  written,  are  marked  (  -1-);  those  lower,  by 
(-).  In  this  case  there  is  no  commensurable  ratio  between  the  prime  t(me  and 
the  other  tones,  that  is,  none  expressible  in  whole  numbers. 

Strew  a  very  thin  membrane  of  this  kind  with  sand,  and  somid  its  prime  tone 
strongly  in  its  neighbourhood  ;  the  sand  will  be  driven  by  the  vibrations  towards  IT 
the  edge,  where  it  collects.  On  producing  another  of  the  tones  of  the  membrane, 
the  sand  collects  in  the  corresponding  nodal  lines,  and  we  are  thus  easily  able  to 
determine  to  which  of  its  tones  the  membrane  has  responded.  A  singer  wdio 
knows  how  to  hit  the  tones  of   the   membrane  correctly,  can  thus  easily  make  the 

spoken  of  in  the  text  (as  in  this  table),  they 
must  be  considered  as  additions  by  the  transla- 
tor. In  the  present  case,  they  give  the  inter- 
vals exactly,  and  not  roughly  as  in  the  column 
of  notes.  Thus,  1439  cents  is  sharper  than  14 
Semitones  above  c,  that  is,  sharper  than  d'  by 
39  hundredths  of  a  Semitone,  or  about  ^  of  a 
Semitone,  and  1858  is  flatter  than  19  Semitones 
above  c,  that  is  flatter  than  g'  by  42  hun- 
dredths of  a  Semitone,  or  nearly  i  a  Semitone. 
— Translator.'] 


*  [Cents  are  hundredths  of  iiu  equal  Semi- 
tone, and  are  exceedingly  valuable  as  measures 
of  any,  especially  unusual,  musical  intervals. 
They  are  fully  exf)lained,  and  the  method  of 
calculating  them  from  the  Interval  Ratios  is 
given  in  App.  XX.  sect.  C.  Here  it  need  only 
be  said  that  the  number  of  hundreds  of  cents 
is  the  number  of  equal,  that  is,  pianoforte 
Semitones  in  the  interval,  and  these  may  be 
counted  on  the  keys  of  any  piano,  while  the 
units  and  tens  show  the  number  of  hundredths 
of  a  Semitone  in  excess.     Wherever  cents  are 


42        INFLUENCE  OF  PART  I ALS  ON  SYMPATHETIC  RESONANCE,     i'aht  i. 

sand  {irranged  itself  at  pleasure  in  one  order  or  the  other,  by  singing  the  correspond- 
ing tones  powerfully  at  a  distance.  But  in  general  the  simpler  figures  of  the  deeper 
tones  are  more  easily  generated  than  the  complicated  figures  of  the  upper  tones. 
It  is  easiest  of  all  to  set  the  membrane  in  general  motion  by  sounding  its  prime 
tone,  and  hence  such  memln-anes  have  been  much  nsed  in  aconstics  to  prove  the 
existence  of  some  determinate  tone  in  some  determinate  spot  of  the  siuTounding 
air.  It  is  most  suitable  for  this  pui-j)ose  to  connect  the  membrane  with  an  inclosed 
mass  of  air.     A,  fig.  15,  is  a  glass  bottle,  Fig.  is. 

having  an  open  month  a,  and  in  place 
of  its  bottom  b,  a  stretched  membrane, 
consisting  of  Avet  pig's  bladder,  al- 
lowed to  dry  after  it  has  been  stretched 
and  fastened.  At  c  is  attached  a 
H  single  fibre  of  a  silk  cocoon,  bearing  a 
drop  of  sealing-wax,  and  hanging  down 
like  a  pendulum  against  the  membrane. 

As  soon  as  the  membrane  vibrates,  the  little  pendulum  is  violently  agitated.  Such 
a  pendulum  is  very  convenient  as  long  as  we  have  no  reason  to  apprehend  any  con- 
fusion of  the  prime  tone  of  the  membrane  with  any  other  of  its  proper  tones.  There 
is  no  scattering  of  sand,  and  the  apparatus  is  therefore  always  in  order.  But  to  decide 
with  certainty  what  tones  are  really  agitating  the  membrane,  we  must  after  all 
place  the  bottle  with  its  mouth  downwards  and  strew  sand  on  the  membrane. 
However,  when  the  bottle  is  of  the  right  size,  and  the  membrane  uniformly 
stretched  and  fastened,  it  is  only  the  prime  tone  of  the  membrane  (slightly  altered 
by  that  of  the  sympathetically  vibrating  mass  of  air  in  the  bottle)  which  is  easily 
excited.  This  prime  tone  can  be  made  deeper  by  increasing  the  size  of  the  mem- 
brane, or  the  volume  of  the  bottle,  or  by  diminishing  the  tension  of  the  membrane 
"  or  size  of  the  orifice  of  the  bottle. 

A  stretched  membrane  of  this  kind,  whether  it  is  or  is  not  attached  to  tie  bot- 
tom of  a  bottle,  will  not  only  be  set  in  vibration  by  nuisical  tones  of  the  same  pitch 
as  its  own  proper  tone,  but  also  by  such  musical  tones  as  contain  the  proper  tone 
of  the  membrane  among  its  upper  partial  tones.  Generally,  given  a  number  of 
interlacing  waves,  to  discover  whether  the  membrane  will  vibrate  sympathetically, 
we  must  suppose  the  motion  of  the  air  at  the  given  place  to  be  mathematically 
analysed  into  a  sum  of  pendular  vibrations.  If  there  is  one  such  vibration  among 
them,  of  which  the  periodic  time  is  the  same  as  that  of  any  one  of  the  proper  tones 
of  the  membrane,  the  corresponding  vibrational  form  of  the  membrane  will  be  super- 
induced. But  if  there  are  none  such,  or  none  sutfieiently  powerful,  the  membrane 
will  remain  at  rest. 

In  this  case,  then,  we  also  find  that  the  analysis  of  the  motion  of  the  air  into 
pendular  vibrations,  and  the  existence  of  certain  vibrations  of  this  kind,  are  deci- 
%  sive  for  the  sympathetic  vibration  of  the  membrane,  and  for  this  purpose  no  other 
similar  analysis  of  the  motion  of  the  air  can  be  substituted  for  its  analysis  into 
pendular  vibrations.  The  pendular  vibrations  into  wdiich  the  composite  motion  of 
the  air  can  be  analysed,  here  show  themselves  capable  of  producing  mechanical 
eflfects  in  external  nature,  independently  of  the  ear,  and  independently  of  mathe- 
matical theory.  Hence  the  statement  is  confirmed,  that  the  theoretical  view  which 
first  led  mathematicians  to  this  method  of  analysing  compound  vibrations,  is 
founded  in  the  nature  of  the  thing  itself. 

As  an  example  take  the  following  descri})tiou  of  a  single  experiment  : — 
A  bottle  of  the  shape  shown  in  fig.   15  above  Avas  covered  with  a  thin  vulcan- 
ised  india-rubber  membrane,   of  which  the  vibrating  surface  was    -49    millimetres 
(1-93  inches)"  in  diameter,  the  bottle  being  UO  millimetres  (5-51   inches)  high,  and 

*  [As  10  inches  are  exactly  25J:  millimetres  the  calculation  of  one  set  of  measures  from 
and  100  metres,  that  is,  100,000  millimetres  are  the  other.  Roughly  we  may  assume  25  mm. 
3937  inches,  it  is  easy  to  form  little  tables  for       to  be  1  inch.     But  wlieuever  dimensions  are 


II 


CHAP.   HI.  RESONATOllS.  43 

having-  an  opening  at  the  brass  month  of  13  milUmetres  (-51  inches)  in  diameter. 
When  blown  it  gave./'ji,  and  the  sand  heaped  itself  in  a  circle  near  the  edge  of  the 
meml)rane.  The  same  circle  resnlted  from  my  giving  the  some  tone  /'jjl  on  an 
harmonium,  or  its  deeper  Octave  /|,  or  the  deeper  Twelfth  B.  Both  Fk  and  D 
gave  the  same  circle,  but  more  weakly.  Now  the  f^  of  the  membrane  is  the  prime 
tone  of  the  harmonium  tone  /"|,  the  second  partial  tone  of  f^,  the  third  of  B,  the 
fourth  of  F^  and  fifth  of  D*  All  these  notes  on  being  sounded  set  the  membrane 
in  the  motion  due  to  its  deepest  tone.  A  second  smaller  circle,  19  millimetres 
(•75  inches)  in  diameter  was  produced  on  the  membrane  by  //  and  the  same  more 
faintly  by  A,  and  there  was  a  trace  of  it  for  the  deeper  Twelfth  e,  that  is,  for  simple 
tones  of  which  vibrational  numbers  were  h  and  i  that  of  !>' .\ 

Stretched  membranes  of  this  kind  are  very  convenient  for  these  and  similar 
experiments  on  the  ])artials  of  compound  tones.     They  have  the  great  advantage 

of  being  independent  of  the  ear,  but  they  H 
are  not  xevy  sensitive  for  the  fainter  simple 
tones.     Their  sensitiveness  is  far  inferior  to 
that  of  the  re^'o^uitors  which  I  have  intro- 
duced.     These  are  hollow  spheres  of  glass 
or   metal,   or  tubes,    with    two  openings   as 
shown  in  figs.  16  a  and  16  b.     One  opening 
(a)  has  sharp  edges,  the  other  (b)  is  funnel- 
shaped,  and  adapted  for  insertion  into  the 
ear.     This  smaller  end  I  usually  coat  with 
melted  sealing  wax,  and  when  the  wax  has 
cooled  down  enough  i;ot  to  hurt  the  finger 
on  being  touchdl,  but  is  still  soft,  I  press  the  opening  into  the  entrance  of  my 
ear.     The  sealing  wax  thus  moulds  itself  to  the  shape  of  the  inner  surface  of  this 
opening,  and  when  I  subsequently  use  the  resonator,  it  fits  easily  and  is  air-tight.  H 
Such  an  instriunont   is  very  like   the  resonance   bottle  already  described,  fig.   15 

(p.  4.2a),  for  which  the  observer's 
Fig.  ic.  1).  VF  ;> 

own    tympanic    membrane    has 

been   made   to  replace   the  for- 
mer artificial  membrane. 

The  mass  of  air  in  a  reso- 
nator, together  with  that  in  the 
aural  passage,  and  with  the 
tympanic  membrane  or  drumskin  itself,  forms  an  elastic  system  which  is  capal)le 
of  vibrating  in  a  peculiar  manner,  and,  in  especial,  the  prime  tone  of  the  sphere, 
which  is  much  deeper  than  any  other  of  its  proper  tones,  can  be  set  into  very 
powerful  sympathetic  vibration,  and  then  the  ear,  which  is  in  immediate  connec- 
tion with  the  air  inside  the  sphere,  perceives  this  augmented  tone  by  direct  action. 
If  we  stop  one  ear  (which  is  best  done  by  a  plug  of  sealing  wax  moulded  into  the  ^ 
form  of  the  entrance  of  the  ear),  J  and  apply  a  resonator  to  the  other,  most  of  the 
tones  produced  in  the  surrounding  air  will  be  considerably  damped ;  but  if  the 
proper  tone  of  the  resonator  is  sounded,  it  brays  into  the  ear  most  powerfully. 

given  inthetext  in  mm.  (that  is,  millimetves),  +  [For    ordinary    purposes    this    is    quite 

thev  will  be  reduced  to  inches  and  decimals  of  enough,  indeed  it  is  generally  unnecessary  to 

an  inch..—rranslatur.]  stop  the  other  ear  at  all.     But  for  such  experi- 

*  [As   the   instrument   was   tempered,   wo  ments  as  Mr.  Bosanquet  had  to  make  on  beats 

should  have,  approximatelv,  for  fjt  the  partials  (see  App.   XX.   section  L.   art.  4,  b)  he  was 

f%  ft,  &c.  ;  for  B  the  partials  'K,  h,  f'%.,  &c. ;  obliged  to  use  a  jar  as  the  resonator,  conduct 

'for>Vthe  partials  i^i,  ft,  4^/%  &c^;  and  the  sound  from  it  through  first  a  glass  and 

for  Z*  the   partials   IJ,  ii,  a,  U',/^,  &c.     To  then  an  elastic  tube  to  a  semicircular  metal  tube 

prevent  confusion  I  have  reduced  the  upper  which  reached  from  ear  to  ear,  to  each  end  of 

partials  of  the  text  to  ordinary  partials,  as  which  a  tube  coated  with  india-rubber,  could  be 

suggested  in  p.  23//,  note.— Translator.]  screwed  into  the  ear.     By  this  means,  when 

t  [Here  the  partials  of  b  are  /),  b',  Sec,  and  proper  care  was   taken,   all   sound   but   that 

of  c  are  c,   c! ,  //,  &c.,  so  that  both  b  and  r.  coming  from  the  resonance  jar  was  perfectly 

contain  b'.— Translator.']  cyicXwAeA.— Translator.'] 


44  RESONATORS.  vmvv  i. 

Hence  any  one,  even  if  he  has  no  ear  for  music  or  is  quite  unpractised  in  detecting 
musical  somids,  is  put  in  a  condition  to  pick  the  required  simple  tone,  even  if  com- 
paratively faint,  from  out  of  a  great  number  of  others.  The  proper  tone  of  the 
resonator  may  even  be  sometimes  heard  cropping  up  in  the  whistling  of  the  wind, 
the  rattling  of  carnage  Avheels,  the  splashing  of  water.  For  these  purposes  such 
resonators  are  incomparably  more  sensitive  than  tuned  membranes.  When  the 
simple  tone  to  be  observed  is  faint  in  comparison  with  those  Avhich  accompany  it, 
it  is  of  advantage  to  alternately  apply  and  withdraw  the  resonator.  We  thus  easily 
feel  whether  the  proper  tone  of  the  resonator  begins  to  sound  when  the  instrument 
is  applied,  whereas  a  unifoi-m  continuous  tone  is  not  so  readily  perceived. 

A  properly  tuned  series  of  such  resonators  is  therefore  an  important  instrument 
for  experiments  in  which  individiial  faint  tones  have  to  be  distinctly  heard,  although 
accompanied  by  others  which  are  strong,  as  in  observations  on  the  combinational 

^  and  Tipper  partial  tones,  and  a  series  of  other  phenomena  to  be  hereafter  described 
relating  to  chords.  By  their  means  such  researches  can  be  carried  out  even  by 
ears  quite  untrained  in  musical  observation,  whereas  it  had  been  previously 
impossible  to  conduct  them  except  by  trained  musical  ears,  and  much  strained 
attention  properly  assisted.  These  tones  were  consequently  accessible  to  the 
observation  of  only  a  very  few  individuals  ;  and  indeed  a  large  number  of  physi- 
cists and  even  musicians  had  never  succeeded  in  distinguishing  them.  And  again 
even  the  trained  ear  is  now  able,  with  the  assistance  of  resonators,  to  carry  the 
analysis  of  a  mass  of  miisical  tones  much  further  than  before.  Without  their  help, 
indeed,  1  shou.ld  scarcely  have  succeeded  in  making  the  observations  hereafter 
described,  with  so  much  precision  and  certainty,  as  I  have  been  enabled  to  attain 
at  present.* 

It  must  be  carefully  noted  that  the  ear  does  not  hear  the  required  tone  with 
augmented  force,  unless  that  tone  attains  a  considerable  intensity  within  the  mass 

H  of  air  enclosed  in  the  resonator.  Now  the  mathematical  theory  of  the  motion  of 
the  air  shows  that,  so  long  as  the  amplitude  of  the  vibrations  is  sufficiently  small, 
the  enclosed  air  will  execute  pendular  oscillations  of  the  same  periodic  time  as 
those  in  the  external  air,  and  none  other,  and  that  only  those  pendular  oscillations 
whose  periodic  time  corresponds  with  that  of  the  proper  tone  of  the  resonator, 
have  any  considerable  strength  ;  the  intensity  of  the  rest  diminishing  as  the  difTer- 
ence  of  their  pitch  from  that  of  the  proper  tone  increases.  All  this  is  independent 
of  the  connection  of  the  ear  and  resonator,  except  in  so  far  as  its  tympanic  mem- 
brane forms  one  of  the  inclosing  walls  of  the  mass  of  air.  Theoretically  this 
apparatus  does  not  differ  from  the  bottle  with  an  elastic  membrane,  in  fig.  15 
(p.  42a),  but  its  sensitiveness  is  amazingly  increased  by  using  the  drumskin  of  the  ear 
for  the  closing  membrane  of  the  bottle,  and  thus  bringing  it  in  direct  connection 
with  the  auditory  nerves  themselves.  Hence  we  cannot  obtain  a  powerful  tone  in 
the   resonator   except   when  an   analysis  of   the   motion  of   the   external   air  into 

^  pendular  vibrations,  would  show  that  one  of  them  has  the  same  periodic  time  as 
the  proper  tone  of  the  resonator.  Here  again  no  other  analysis  but  that  into 
pendular  vibrations  woidd  give  a  correct  result. 

It  is  easy,  for  an  observer  to  convince  himself  of  the  above-named  properties  of 
resonators.  Apply  one  to  the  ear,  and  let  a  piece  of  harmonised  miisic,  in  which 
the  proper  tone  of  the  resonator  frequently  occurs,  be  executed  by  any  instruments. 
As  often  as  this  tone  is  struck,  the  ear  to  which  the  instrument  is  held,  will  hear 
it  violently  contrast  with  all  the  other  tones  of  the  chord. 

This  proper  tone  will  also  often  be  heard,  but  more  weakly,  when  deeper 
musical  tones  occur,  and  on  investigation  we  find  that  in  such  cases  tones  have 
been  struck  which  include  the  proper  tone  of  the  resonator  among  their  upper 
partial  tones.  Such  deeper  musical  tones  are  called  the  harmonic  nnder  tones  of 
the  resonator.  They  are  musical  tones  whose  periodic  time  is  exactly  2,  3,  4,  5, 
and  so  on,  times  as  great  as  that  of  the  resonator.     Thus  if  the  proper  tone  of 

*  See  Appendix  II.  for  the  measures  and  different  foi'ins  of  these  Resonators. 


CHAP.  iir. 


SYMPATHETIC  RESONANCE  OF  STRINGS.  45 


the  resonator  is  c",  it  will  be  heard  when  a  nmsical  instruuieut  sounds  r', ./",  c,  A\y, 
F,  D,  C,  and  so  on.*  In  this  case  the  resonator  is  made  to  sound  in  sympathy 
with  one  of  the  harmonic  xipper  partial  tones  of  the  compound  nmsical  tone  which 
is  vibrating  in  the  external  air.  It  must,  however,  be  noted  that  by  no  means  all 
the  harmonic  upper  partial  tones  occur  in  the  compound  tones  of  every  instrument, 
and  that  they  have  very  difterent  degrees  of  intensity  in  different  instruments.  In 
tlie  musical  tones  of  violins,  pianofortes,  and  harmoniums,  the  first  five  or  six  are 
generally  very  distinctly  present.  A  more  detailed  account  of  the  iipper  partial 
tones  of  strings  will  be  given  in  the  next  chapter.  On  the  harmonium  the  un- 
evenly numbered  partial  tones  (1,  3,  5,  &c.)  are  generally  stronger  than  the  evenly 
numbered  ones  (2,  4,  6,  &c.).  In  the  same  way,  the  upper  partial  tones  are  clearly 
heard  by  means  of  the  resonators  in  the  singing  tones  of  the  luunan  voice,  but 
differ  in  strength  for  the  different  vowels,  as  will  be  shown  hereafter.  11 


Among  the  bodies  capable  of  strong  sympathetic  vibration  must  be  reckoned 
stretched  strings  which  are  connected  with  a  sounding  board,  as  on  the  pianoforte. 

The  principal  mark  of  distinction  between  strings  and  the  other  bodies  which 
vibrate  sympathetically,  is  that  different  vibrating  forms  of  strings  give  simple 
tones  corresponding  to  the  htwmonic  upper  partial  tones  of  the  prime  tone,  whereas 
the  secondary  simple  tones  of  membranes,  bells,  rods,  &c.,  are  wdiarmonic  with  the 
prime  tone,  and  the  masses  of  air  in  resonators  have  generally  only  very  high 
upper  partial  tones,  also  chiefly  «?iharmonic  with  the  prime  tone,  and  not  capalde 
of  being  much  reinforced  by  the  resonator. 

The  vibrations  of  strings  may  be  studied  either  on  elastic  chords  loosely 
stretched,  and  not  sonorous,  but  swinging  so  slowly  that  their  motion  may  be 
followed  with  the  hand  and  eye,  or  else  on  sonorous  strings,  as  those  of  the  piano- 
forte, guitar,  monochord,  or  violin.  Strings  of  the  first  kind  are  best  made  of  thin  H 
spirals  of  brass  wire,  six  to  ten  feet  in  length.  They  should  be  gently  stretched, 
and  both  ends  should  be  fastened.  A  string  of  this  construction  is  capable  of 
making  very  large  excursions  with  great  regularity,  which  are  easily  seen  by  a  large 
audience.  The  swings  are  excited  by  moving  the  string  regularly  backwards  and 
forwards  by  the  finger  near  to  one  of  its  extremities. 

A  string  may  be  first  made  to  vibrate  as  in  fig.  17,  a  (p.  466),  so  that  its  appear- 
ance when  displaced  from  its  position  of  rest  is  always  that  of  a  simple  half  wave. 
The  string  in  this  case  gives  a  single  simple  tone,  the  deepest  it  can  produce,  and 
no  other  harmonic  secondary  tones  are  audible. 

But  the  string  may  also  during  its  motion  assume  the  forms  fig.  17,  b,  c,  d. 
In  this  case  the  form  of  the  string  is  that  of  two,  three,  or  four  half  waves  of  a 
simple  wave-curve.  In  the  vibrational  form  b  the  string  produces  only  the  upper 
Octave  of  its  prime  tone,  in  the  form  c  the  Twelfth,  and  in  the  form  d  the  second 
Octave.  The  dotted  lines  show  the  position  of  the  string  at  the  end  of  half  its  11 
periodic  time.  In  b  the  point  ft  remains  at  rest,  in  c  two  points  yi  and  y.  remain 
at  rest,  in  d  three  points  Sj,  8.,,  &,.  These  points  are  called  nodes.  In  a  swinging 
spiral  wire  the  nodes  are  readily  seen,  and  for  a  resonant  string  they  are  shown  by 
little  paper  riders,  which  are  jerked  off  from  the  vibrating  parts  and  remain  sitting 
on  the  nodes.  When,  then,  the  string  is  divided  by  a  node  into  two  swinging 
sections,  it  produces  a  simple  tone  having  a  pitch  number  double  that  of  the  prime 

*  [The  (.•"  occurs  as  the  2nd,  3rd,  4th,  the  7th  being  rather  flat.  The  partials  are 
5th,   Gth,    7th,    8th    partials   of   these   notes,       in  fact  :— 

c'      c" 

f      f      <-■" 

c    c'  r  c" 

A\y  fAJj  c\)  a'\)  c" 

F  f  c'  f  a'  c" 

I)  d  a  d'  ft  a.       c" 

a  c  f  c  e'  /'      h'V,    c".-- Translator.^ 


46 


SYMPATHETIC  RESOXANCE  OF  STRINGS. 


tone.     For  three  sections  the  pitch  number  is  tripled,  for  foiu-  sections  iiu;i(h-ui)led, 
and  so  on. 

To  bring  a  spiral  wire  into  these  different  forms  of  vibration,  we  move  it 
periodically  with  the  finger  near  one  extremity,  adopting  the  period  of  its  slowest 
swings  for  a,  twice  that  rate  for  b,  three  times  for  c,  and  four  times  for  d.  Or  else 
we  just  gently  touch  one  of  the  nodes  nearest  the  extremity  with  the  finger,  and  jjluck 
the  string  half-way  between  this  node  and  the  nearest  end.  Hence  when  yi  in  c, 
or  81  in  d,  is  kept  at  rest  by  the  finger,  we  pluck  the  string  at  c.  Tlie  other  nodes 
then  appear  when  the  vibrati(jn  commences. 


For  a  sonorous  string  the  vibrational  forms  of  fig.  17  above  arc  most  purely 
produced  by  applying  to  its  sounding  board  the  handle  of  a  tuning-fork  which  has 
been  struck  and  gives  the  simple  tone  corresponding  to  the  form  required.  If  only 
a  determinate  number  of  nodes  are  desired,  and  it  is  indifferent  whether  the  indi- 
vidual points  of  the  string  do  or  do  not  execute  simple  vibrations,  it  is  sufticient  to 
touch  the  string  very  gently  at  one  of  the  nodes  and  either  pluck  the  string  or  rub 
it  with  a  violin  bow.  By  touching  the  string  with  the  finger  all  those  simple  vibra- 
tions are  damped  which  have  no  node  at  that  point,  and  only  those  remain  which 
allow  the  string  to  be  at  rest  in  that  place. 

The  number  of  nodes  in  long  thin  strings  may  be  considerable.  They  cease  to 
be  formed  when  the  sections  which  lie  between  the  nodes  are  too  short  and  stiff  to 
U  be  capable  of  sonorous  vibration.  Very  fine  strings  consequently  give  a  greater 
number  of  higher  tones  than  thicker  ones.  On  the  violin  and  the  lower  pianoforte 
strings  it  is  not  very  difficult  to  produce  tones  with  10  sections;  but  with  extremelv 
fine  wires  tones  with  16  or  20  sections  can  be  made  to  sound.    [Also  compare  p.  7Sd.] 

The  forms  of  vibration  here  spoken  of  are  those  in  which  each  point  of  the 
string  performs  pendular  oscillations.  Hence  these  motions  excite  in  the  ear  the 
sensation  of  only  a  single  simple  tone.  In  all  other  vibratiomxl  forms  of  the 
strings,  the  oscillations  are  not  simply  pendular,  but  take  place  according  to  a  differ- 
ent and  more  complicated  law.  This  is  always  the  case  when  the  string  is  plucked 
in  the  usual  way  with  the  finger  (as  for  guitar,  harp,  zither)  or  is  struck  with  a 
hammer  (as  on  the  pianoforte),  or  is  rubbed  with  a  violin  bow.  The  resulting  motions 
may  then  be  regarded  as  compounded  of  many  simple  vibrations,  which,  when 
taken  separately,  correspond  to  those  in  fig.  17.  The  multiplicity  of  such  com- 
posite foi-ms  of  motion  is  infinitely  great,  the  string  may  indeed  be  considered 
as  capable  of  assuming  any  given  form  (provided  we  confine  ourselves  in  all  cases 


CHAP.  III.  SYMPATHETIC  KESOXANCK  ()E  .STRINGS.  4  7 

to  very  small  deviations  from  the  position  of  rest),  because,  according-  to  wiiat  was 
said  in  Chapter  II.,  any  given  form  of  wave  can  he  compounded  ont  of  a  number 
of  simple  waves  such  as  those  indicated  in  tig.  17,  a,  b,  c,  d.  A  plucked,  struck, 
or  bowed  string  therefore  allows  a  great  number  of  harmonic  upper  partial  tones  to 
l)e  heard  at  the  same  time  as  the  prime  tone,  and  generally  the  number  increases 
with  the  thinness  of  the  string.  The  peculiar  tinkling  sound  of  very  fine  metallic 
.strings  is  clearly  due  to  these  very  high  secondary  tones.  It  is  easy  to  distinguish 
the  upper  simple  tones  up  to  the  sixteenth  by  means  of  resonators.  Beyond  the 
sixteenth  they  are  too  close  to  each  other  to  be  distinctly  separable  by  this  mean.s. 

Hence  when  a  string  is  sympathetically  excited  by  a  musical  tone  in  its  neigh- 
])ourhood,  answering  to  the  i)itch  of  the  prime  tone  of  the  string,  a  whole  series  of 
difterent  simple  vibrational  forms  will  generally  be  at  the  same  time  generated  in 
the  string.  For  when  the  prime  of  the  musical  tone  corresponds  to  the  pi'ime  of 
the  string  all  the  harmonic  upper  partials  of  the  first  correspond  to  those  of  the  H 
second,  and  are  hence  capable  of  exciting  the  corresponding  vibrational  forms  in 
the  string.  Generally  the  string  will  be  brought  into  as  many  forms  of  sympa- 
thetic vibration  by  the  motion  of  the  air,  as  the  analysis  of  that  motion  shows  that 
it  possesses  simple  vibrational  forms,  having  a  periodic  time  equal  to  that  of  some 
vibrational  form,  that  the  string  is  capable  of  assuming.  But  as  a  general  rule 
when  there  is  one  such  simple  vibrational  form  in  the  air,  there  are  several  such, 
and  it  will  often  be  difficult  to  determine  by  which  one,  out  of  the  many  possible 
simple  tones  which  would  produce  the  effect,  the  string  has  been  excited.  Conse- 
(|uently  the  usual  unweighted  strings  are  not  so  convenient  for  the  determination 
of  the  pitch  of  any  simple  tones  which  exist  in  a  composite  mass  of  air,  as  the 
membranes  or  the  inclosed  air  of  resonators. 

To  make  experiments  with  the  pianoforte  on  the  sympathetic  vibrations  of 
strings,  select  a  flat  instrument,  raise  its  lid  so  as  to  expose  the  strings,  then  press 
down  the  key  of  the  string  (for  c'  suppose)  wdiich  you  wish  to  put  into  sympathetic 
vibration,  but  so  slowly  that  the  hammer  does  not  strike,  and  place  a  little  chip  of  ^ 
wood  across  this  c  string.  You  will  find  the  chip  put  in  motion,  or  even  thrown 
otf,  when  certain  other  strings  are  struck.  The  motion  of  the  chip  is  greatest  when 
one  of  the  under  tones  of  c  (p.  iid)  is  struck,  as  c,  F,  C,  A}),  F,  D^,  or  C ^.  Some, 
but  much  less,  motion  also  occurs  when  one  of  the  upper  partial  tones  of  c  is 
struck,  as  c",  g" ,  or  c'",  but  in  this  last  case  the  chip  wdll  not  move  if  it  has  been 
placed  over  one  of  the  corresponding  nodes  of  the  string.  Thus  if  it  is  laid  across 
the  middle  of  the  string  it  will  be  still  for  c"  and  c",  but  will  move  for  g" .  Placed 
at  one  third  the  length  of  the  string  from  its  extremity,  it  will  not  stir  for  (/",  but 
will  move  for  c"  or  c" .  Finally  the  string  c  will  also  be  put  in  motion  when  an 
under  tone  of  one  of  its  upper  partial  tones  is  struck;  for  example,  the  note/,  of  which 
the  third  partial  tone  c"  is  identical  with  the  second  partial  tone  of  c'.  In  this  case 
also  the  chip  remains  at  rest  when  put  on  to  the  middle  of  the  string  /,  which  is 
its  node  for  c".  In  the  same  way  the  string  c  will  move,  with  the  formation  of  H 
two  nodes,  for  g ,  g,  or  e\f,  all  which  notes  have  (/"  as  an  upper  partial  tone,  which 
is  also  the  third  partial  of  c  r' 

Observe  that  on  the  pianoforte,  when  one  end  of  the  strings  is  commonly 
concealed,  the  position  of  the  nodes  is  easily  found  by  pressing  the  string  gently 
on  both  sides  and  striking  the  key.  If  the  finger  is  at  a  node  the  corresponding 
upper  partial  tone  will  be  heard  purely  and  distinctly,  otherwise  the  tone  of  the 
string  is  dull  and  bad. 

As  long  as  only  one  upper  partial  tone  of  the  string  c'  is  excited,  the  corre- 
sponding nodes  can  be  discovei-ed,  and  hence  the  particular  form  of  its  vibration 
determined.     But  this  is  no  longer  possible  by  the  above  mechanical  method  when 

*  [These  experiments  can  of  course  not  be  struck  and  damped.     And  this  sounding  of  c', 

conducted  on  the  usual  upright  cottage  piano.  although  unstruck,  is  itself  a  very  interesting 

But  the  experimenter  can  at  least  hear  the  phenomenon.     But  of  course,  as  it  depends  on 

tone   of   f',    if   c,    F,   C,   &c.,  are  struck   and  the  ear,  it  does  not  establish  the  results  of  the 

immediately   damped,    or    if    c",    ij" ,   c'"   are  text. —  Translator.'] 


48  OBJECTIVE  EXISTENCE  OV  PAIITLVLS.  part  i. 

two  upper  partial  tones  are  excited,  such  ;is  r"  and  ;/",  as  would  l)e  the  case  if  both 
these  notes  were  struck  at  once  on  the  ijianoforte,  because  the  whole  string  of  r' 
would  then  be  in  motion. 

Although  the  relations  for  strings  appear  more  complicated  to  the  eye,  their 
sympathetic  vibration  is  subject  to  the  same  law  as  that  which  holds  for  resonators, 
membranes,  and  othei-  elastic  bodies.  The  sympathetic  vibration  is  always  deter- 
mined by  the  analysis  of  whatever  sonorous  motions  exist,  into  simple  pendidar 
vibrations.  If  the  periodic  time  of  one  of  these  simple  vibrations  corresponds  to 
the  periodic  time  of  one  of  the  proper  tones  of  the  elastic  body,  that  body,  whether 
it  be  a  string,  a  membrane,  or  a  mass  of  air,  will  be  put  into  strong  sympathetic 
vibration. 

These  facts  give  a  real  objective  value  to  the  analysis  of  sonorous  motion  into 
simple  pendular  vibration,  and  no  such  vahie  would  attach  to  any  other  analysis. 
H  Every  individual  single  system  of  waves  formed  by  pendular  vibrations  exists  as 
an  independent  mechanical  unit,  expands,  and  sets  in  motion  other  elastic  bodies 
having  the  corresponding  proper  tone,  perfectly  undisturbed  by  any  other  simple 
tones  of  other  pitches  which  may  be  expanding  at  the  same  time,  and  Avhich  may 
proceed  either  from  the  same  or  any  other  source  of  sound.  Each  single  simple 
tone,  then,  can,  as  we  have  seen,  be  separated  from  the  composite  mass  of  tones, 
by  mechanical  means,  namely  by  bodies  which  will  vibrate  sympathetically  with 
it.  Hence  every  individual  partial  tone  exists  in  the  compound  musical  tone 
produced  by  a  single  musical  instrument,  just  as  truly,  and  in  the  same  sense,  as  the 
different  colours  of  the  rainbow  exist  in  the  white  light  proceeding  from  the  sun 
or  any  other  luminous  body.  Light  is  also  oidy  a  vibrational  motion  of  a  peculiar 
elastic  medium,  the  luminous  ether,  just  as  sound  is  a  vibrational  motion  of  the 
air.  In  a  beam  of  Avhite  light  there  is  a  species  of  motion  which  /nai/  be  repre- 
sented as  the  sum  of  many  oscillatory  motions  of  various  periodic  times,  each  of 
H  which  corresponds  to  one  particular  colour  of  the  solar  spectrum.  But  of  course 
each  particle  of  ether  at  any  particular  moment  has  only  one  determinate  velocity, 
and  only  one  determinate  departvire  from  its  mean  position,  just  like  each  particle 
of  air  in  a  space  traversed  by  many  systems  of  sonorous  waves.  The  really  exist- 
ing motion  of  any  particle  of  ether  is  of  course  only  one  and  indi^'idual ;  and  our 
theoretical  treatment  of  it  as  compound,  is  in  a  certain  sense  arbitrary.  But  the 
imdulatory  motion  of  light  can  also  be  analysed  into  the  waves  corresponding  to 
the  separate  colours,  by  external  mechanical  means,  such  as  by  refraction  in  a 
prism,  or  by  transmission  through  fine  gratings,  and  each  individual  simple  wave 
of  light  corresponding  to  a  simple  colour,  exists  mechanically  by  itself,  indepen- 
dently of  any  other  colour. 

We  must  therefore  not  liold  it  to  be  an  illusion  of  the  ear,  or  to  be  mere 
imagination,  when  in  the  musical  tone  of  a  single  note  emanating  from  a  musical 
instrument,  we  distinguish  many  partial  tones,  as  I  have  found  musicians  inclined 
^  to  think,  even  when  they  have  heard  those  partial  tones  quite  distinctly  with  their 
own  ears.  If  we  admitted  this,  we  should  have  also  to  look  upon  the  colours  of 
the  spectrum  which  are  separated  from  white  light,  as  a  mere  illusion  of  the  eye. 
The  real  outward  existence  of  partial  tones  in  nature  can  be  established  at  any 
moment  by  a  sympathetically  vibrating  membrane  which  casts  up  the  sand  strewn 
upon  it. 

Finally  I  would  observe  that,  as  respects  the  conditions  of  sympathetic  vibra- 
ti(jn,  I  have  been  obliged  to  refer  frequently  to  the  mechanical  theory  of  the 
motion  of  air.  Since  in  the  theory  of  soiuid  we  have  to  deal  wuth  well-known 
mechanical  forces,  as  the  pressure  of  the  air,  and  Avith  motions  of  material 
particles,  and  not  with  any  hypothetical  explanation,  theoretical  mechanics  have 
an  unassailable  a\ithority  in  this  department  of  science.  Of  course  those  readers 
who  are  unacquainted  with  mathematics,  must  accept  the  results  on  faith.  An 
experimental  way  of  examining  the  problems  in  question  will  be  described  in  the 
next  chapter,  in  which  the  laws  of  the  analysis  of  musical  tones  by  the  ear  have 


CHAPS.  III.  IV.         METHODS  OF  OBSERVING  PARTIAL  TONES.  49 

to  be  established.  The  experimental  proof  there  given  for  the  ear,  can  also  be 
carried  out  in  precisely  the  same  way  for  membranes  and  masses  of  air  which 
vibrate  sympathetically,  and  the  identity  of  the  laws  in  l)oth  cases  will  result  from 
those  iiwestiffations.* 


CHAPTER  IV. 

ON  THE  ANALYSIS  OF  MUSICAL  TONES  BY  THE  EAR. 

It  was  frequently  mentioned  in  the  preceding  chapter  that  musical  tones  could  be 
resolved  by  the  ear  alone  unassisted  by  any  peculiar  apparatus,  into  a  series  of 
partial  tones  corresponding  to  the  simple  pendular  vibrations  in  a  mass  of  air,  that  ^ 
is,  into  the  same  constituents  as  those  into  which  the  motion  of  the  air  is  resolved 
b}^  the  sympathetic  vibration  of  elastic  bodies.  We  proceed  to  show  the  correctness 
of  this  assertion. 

Any  one  who  endeavours  for  the  first  time  to  distinguish  the  upper  partial 
tones  of  a  musical  tone,  generally  finds  considerable  difficulty  in  merely  hearing 
them. 

The  analysis  of  our  sensations  when  it  cannot  be  attached  to  corresponding 
differences  in  external  objects,  meets  with  peculiar  difficulties,  the  nature  and 
significance  of  which  will  have  to  be  considered  hereafter.  The  attention  of  the 
observer  has  generally  to  be  drawn  to  the  phenomenon  he  has  to  observe,  by 
peculiar  aids  properly  selected,  until  he  knows  precisely  what  to  look  for ;  after  he 
has  once  succeeded,  he  will  be  able  to  throw  aside  such  crutches.  Similar  diffi- 
culties meet  us  in  the  observation  of  the  upper  partials  of  a  musical  tone.  I  shall 
first  give  a  description  of  such  processes  as  will  most  easily  put  an  untrained  H 
observer  into  a  position  to  recognise  upper  partial  tones,  and  I  will  remark  in 
passing  that  a  musically  trained  ear  will  not  necessarily  hear  tipper  partial  tones 
with  greater  ease  and  certainty  than  an  untrained  ear.  Success  depends  rather 
upon  a  peculiar  power  of  mental  abstraction  or  a  peculiar  mastery  over  attention, 
than  upon  musical  training.  But  a  musically  trained  observer  has  an  essential 
advantage  over  one  not  so  trained  in  his  power  of  figuring  to  himself  how  the 
simple  tones  sought  for,  ought  to  sound,  whereas  the  untrained  observer  has  con- 
tinually to  hear  these  tones  sounded  by  other  means  in  order  to  keep  their  effect 
fresh  in  his  mind. 

First  we  must  note,  that  the  unevenly  numbered  partials,  as  the  Fifths,  Thirds, 
Sevenths,  &c.  of  the  prime  tones,  are  usually  easier  to  hear  than  the  even  ones, 
which  are  Octaves  either  of  the  prime  tone  or  of  some  of  the  upper  partials  which 
lie  near  it,  just  as  in  a  chord  we  more  readily  distinguish  whether  it  contains 
Fifths  and  Thirds  than  whether  it  has  Octaves.  The  second,  fourth,  and  eighth  H 
partials  are  higher  Octaves  of  the  prime,  the  sixth  partial  an  Octave  above  the 
third  partial,  that  is,  the  Twelfth  of  the  prime ;  and  some  practice  is  required  for 
distinguishing  these.  Among  the  uneven  partials  which  are  more  easily  dis- 
tinguished, the  first  place  must  be  assigned,  from  its  usual  loudness,  to  the  third 
partial,  the  Twelfth  of  the  prime,  or  the  Fifth  of  its  first  higher  Octave.  Then 
follows  the  fifth  partial  as  the  major  Third  of  the  prime,  and,  generally  verj  faint, 
the  seventh  partial  as  the  minor  Seventh  +  of  the  second  higher  Octave  of  the 
prime,  as  will  be  seen  by  their  following  expression  in  musical  notation,  for  the 
compound  tone  c. 

*  Optical  means  for  rendering  visible  weak  f  [Or  more  correctly  Awiv-miuor  Seventh  ; 

sympathetic  motions  of  sonorous  masses   of  as  the  real  minor  Seventh,  formed  by  taking 

air,  are  described  in  App.  II.     These  means  two  Fifths  down  and  then  two  Octaves  up,  is 

are  valuable  for  demonstrating  the  facts  to  sharper  by  27  cents,  or  in  the  ratio  of  63  :  64. 

hearers  unaccustomed  to  the  observing  and  — Translator. '\ 
distinguishing  musical  tones. 


50  METHODS  OF  OBSERVING  PARTIAL  TONES. 

1 in r^- 


BtE^^EE: 


:p= 


-:& 


I  2         '^  3  4  5^  &^  _7^  «^ 

r  (■'  r/  r"  ^"  v"  '//'[j         c'" 

[Cents.  0  1200  1902        2400        2786        3i02        3369        3600]* 

In  commencing  to  observe  upper  partial  tones,  it  is  advisable  just  before  pro- 
ducing the  musical  tone  itself  which  you  wish  to  analyse,  to  sound  the  note  you 
wish  to  distinguish  in  it,  very  gently,  and  if  possible  in  the  same  quality  of  tone 
as  the  compound  itself.  The  pianoforte  and  harmonivim  are  well  adapted  for 
these  experiments,  because  they  both  have  upper  partial  tones  of  considerable 
power. 

H  First  genth'  strike  on  a  piano  the  note  g',  as  marked  above,  and  after  letting 
the  digital  +  rise  so  as  to  damp  the  string,  strike  the  note  c,  of  which  g'  is  the 
third  partial,  with  great  force,  and  keep  your  attention  directed  to  the  pitch  of  the 
(/'  which  you  had  just  heard,  and  you  will  hear  it  again  in  the  compound  tone  of 
c.  Similarly,  first  stroke  the  fifth  partial  e"  gently,  and  then  c  strongly.  These 
upper  partial  tones  are  often  more  distinct  as  the  sound  dies  away,  because  they 
appear  to  lose  force  more  slowly  than  the  prime.  The  seventh  and  ninth  partials 
h"\f  and  d"  are  mostly  weak,  or  quite  absent  on  modern  pianos.  If  the  same  ex- 
periments are  tried  with  an  harmonium  in  one  of  its  louder  stops,  the  seventh 
partial  will  generally  be  well  heard,  and  sometimes  even  the  ninth. 

To  the  objection  which  is  sometimes  made  that  the  observer  only  imagines  he 
hears  the  partial  tone  in  the  compound,  because  he  had  just  heard  it  by  itself,  I 
need  only  remark  at  present  that  if  e"  is  first  heard  as  a  partial  tone  of  c  on  a 
good  piano,  tuned  in  equal  temperament,  and  then  /'  is  struck  on  the  instrument 

ff  itself,  it  is  quite  easy  to  perceive  that  the  latter  is  a  little  sharper.  This  follows 
from  the  method  of  tuning.  But  if  there  is  a  difference  in  pitch  between  the  two 
tones,  one  is  certainly  not  a  continuation  of  the  mental  effect  produced  by  the 
other.  Other  facts  which  completely  refute  the  above  conception,  will  be  subse- 
quently adduced. 

A  still  more  suitable  process  than  that  just  described  for  the  piano,  can  be 
adopted  on  any  stringed  instrument,  as  the  piano,  monochord,  or  violin.  It  con- 
sists in  first  producing  the  tone  we  wish  to  hear,  as  an  harmonic  [p.  25(7,  note]  by 
touching  the  corresponding  node  of  the  string  when  it  is  struck  or  rubbed.  The 
resembhmce  of  the  tone  first  heard  to  the  corresponding  partial  of  the  compound 
is  then  much  greater,  and  the  ear  discovers  it  more  readily.  It  is  usual  to  place  a 
divided  scale  bj^  the  string  of  a  monochord,  to  facilitate  the  discovery  of  the  nodes. 
Those  for  the  third  partial,  as  shown  in  Chap.  III.  (p.  idd),  divide  the  string  into 
three  equal  parts,  those  for  the  fifth  into  five,  and  so  on.     On  the  piano  and  violin 

U  the  position  of  these  points  is  easily  found  experimentally,  by  touching  the  string 
gently  with  the  finger  in  the  neighbourhood  of  the  node,  which  has  been  approxi- 
matively  detennined  by  the  eye,  then  striking  or  bowing  the  sti'ing,  and  moving 
the  finger  about  till  the  required  harmonic  comes  out  strongly  and  purely.  By 
then  sounding  the  string,  at  one  time  with  the  finger  on  the  node,  and  at  another 
without,  we  obtain  the  required  upper  partial  at  one  time  as  an  harmonic,  and  at 
another  in  the  compound  tone  of  the  whole  string,  and  thus  learn  to  recognise  the 
existence  of  the  first  as  part  of  the  second,  with  comparative  ease.  Using  thin 
strings  which  have  loud  upper  partials,  I  have  thus  been  able  to  recognise  the 

*  [The  cents  (see  p.  ild,  note),  reckoned  piano  or  organ,  are  best  called  digitals  or 
from  the  lowest  note,  are  assigned  on  the  finger-keys,  on  the  analogy  of  pedals  and  foot- 
supposition  that  the  harmonics  are  perfect,  keys  on  the  organ.  The  word  key  ha\-ing 
as  on  the  Harmonical,  not  tempered  as  on  another  musical  sense,  namely,  the  scale  in 
the  pianoforte.  See  also  diagram,  p.  22t:. —  which  a  piece  of  music  is  written,  will  without 
Translator.]  prefix  be  confined  to  this  meaning. — Trans- 

t  [The   keys   played   by  the  fingers   on   a  lalor.] 


CHAP.  IV.  METHODS  OF  0BSP:RVING  PARTIAL  TONP:s.  51 

partials  separately,  up  to  the  sixteenth.     Those  which  He  still  higher  are  too  near 
to  each  other  in  pitch  for  the  ear  to  separate  them  readily. 

In  such  experiments  I  recommend  the  following  process.  Touch  the  node  of 
the  string  on  the  pianoforte  or  monochord  with  a  camel's-hair  pencil,  strike  the 
note,  and  immediately  remove  the  pencil  from  the  string.  If  the  ])encil  has  been 
pressed  tightly  on  the  strisig,  we  either  continue  to  hear  the  required  partial  as  an 
harmonic,  or  else  in  addition  hear  the  prime  tone  gently  sounding  with  it.  On 
repeating  the  excitement  of  the  string,  and  continuing  to  press  more  and  more 
lightly  with  the  camel's-hair  pencil,  and  at  last  removing  the  pencil  entirely,  the 
prime  tone  of  the  string  will  be  heard  more  and  more  distinctly  with  the  harmonic 
till  we  have  finally  the  full  natural  miisical  tone  of  the  string.  By  this  means 
we  obtain  a  series  of  gradual  transitional  stages  between  the  isolated  partial  and 
the  compound  tone,  in  which  the  first  is  readily  retained  by  the  ear.  By  applying 
this  last  process  I  have  generally  succeeded  in  making  perfectly  untrained  ears  H 
recognise  the  existence  of  upper  partial  tones. 

It  is  at  first  more  difticult  to  hear  the  upper  partials  on  most  wind  instruments 
and  in  the  human  voice,  than  on  stringed  instruments,  harmoniums,  and  the  more 
penetrating  stops  of  an  organ,  because  it  is  then  not  so  easy  first  to  produce  the 
upper  partial  softly  in  the  same  quality  of  tone.  But  still  a  little  practice  sufliices 
to  lead  the  ear  to  the  required  partial  tone,  by  previously  touching  it  on  the  piano. 
The  partial  tones  of  the  human  voice  are  comparativel}'  most  difficult  to  distinguish 
for  reasons  which  will  be  given  svibsequently.  Nevertheless  they  were  distin- 
guished even  by  Rameau*  without  the  assistance  of  any  apparatus.  The  process 
is  as  follows  : — 

Get  a  powerful  bass  voice  to  sing  e]^  to  the  vowel  0,  in  sore  [more  like  m',> 
in  sail'  than  o  in  so],  gently  touch  b'\)  on  the  piano,  which  is  the  Twelfth,  or 
third  partial  tone  of  the  note  e\f,  and  let  its  sound  die  away  while  you  ai-e  listening 
to  it  attentively.  The  note  l>'\f  on  the  piano  will  appear  really  not  to  die  away,  H 
but  to  keep  on  sounding,  even  when  the  string  is  damped  by  removing  the  finger 
from  the  digital,  because  the  ear  unconsciously  passes  from  the  tone  of  the  piano 
to  the  partial  tone  of  the  same  pitch  produced  by  the  singer,  and  takes  the  latter 
for  a  continuation  of  the  former.  But  when  the  finger  is  removed  from  the  key, 
and  the  damper  has  fallen,  it  is  of  course  impossible  that  the  tone  of  the  string 
should  have  continued  sounding.  To  make  the  experiment  for  g"  the  fifth  partial, 
or  major  Third  of  the  second  Octave  above  e\),  the  voice  should  sing  to  the  vowel 
A  in  father. 

The  resonators  described  in  the  last  chapter  furnish  an  excellent  means  for 
this  purpose,  and  can  be  used  for  the  tones  of  any  musical  instrument.  On  apply- 
ing to  the  ear  the  resonator  corresponding  to  any  given  upper  partial  of  the  com- 
pound c,  such  as  g',  this  g'  is  rendered  much  more  powerful  when  c  is  sounded. 
Now  hearing  and  distinguishing  g'  in  this  case  by  no  means  proves  that  the  ear 
alone  and  without  this  apparatus  would  hear  g'  as  part  of  the  compound  c.  But  U 
the  increase  of  the  loudness  of  g'  caused  by  the  resonator  may  be  used  to  direct 
the  attention  of  the  ear  to  the  tone  it  is  required  to  distinguish.  On  gradually 
removing  the  resonator  from  the  ear,  the  force  of  g'  will  decrease.  But  the 
attention  once  directed  to  it  by  this  means,  remains  more  readily  fixed  upon 
it,  and  the  observer  continues  to  hear  this  tone  in  the  natural  and  unchanged 
compound  tone  of  the  given  note,  even  with  his  unassisted  ear.  The  sole  office 
of  the  resonators  in  this  case  is  to  direct  the  attention  of  the  ear  to  the  required 
tone. 

By  frequently  instituting  similar  experiments  for  perceiving  the  upjier  partial 
tones,  the  observer  comes  to  discover  them  more  and  more  easily,  till  he  is  finally 
able  to  dispense  with  any  aids.  But  a  certain  amount  of  undisturbed  concentration 
is  always  necessary  for  analysing  musical  tones  by  the  ear  alone,  and  hence  the 
use  of  resonators  is  quite  indispensable  for  an  accurate  comparison  of  different 
*  Nmivemt  SysUme  de  Musique  thioriquc.     Paris :  1726.     Preface. 


52  PROOF   OF  OHM'S  LAW.  part  i. 

qualities  of  tone,  especially  in  respect  to  the  weaker  upper  partials.  At  least,  I 
must  confess,  that  my  own  attempts  to  discover  the  upper  partial  tones  in  the 
human  voice,  and  to  determine  their  differences  for  different  vowels,  were  most 
unsatisfactory  until  I  applied  the  resonators. 

We  now  proceed  to  prove  that  the  human  ear  really  does  analyse  musical 
tones  according  to  the  law  of  simple  vibrations.  Since  it  is  not  possible  to  insti- 
tute an  exact  comparison  of  the  strength  of  our  sensations  for  different  simple 
tones,  we  must  confine  ourselves  to  proving  that  when  an  analysis  of  a  composite 
tone  into  simple  vibrations,  effected  by  theoretic  calculation  or  by  sympathetic 
resonance,  shows  that  certain  upper  partial  tones  are  absent,  the  ear  also  does 
not  perceive  them. 

The  tones  of  strings  are  again  best  adapted  for  conducting  this  proof,  because 
they  admit  of  many  alterations  in  their  quality  of  tone,  according  to  the  manner 
H  and  the  spot  in  which  they  are  excited,  and  also  because  the  theoretic  or  experi- 
mental analysis  is  most  easily  and  completely  performed  for  this  case.  Thomas 
Young*  first  showed  that  when  a  string  is  plucked  or  struck,  or,  as  we  may  add, 
bowed  at  any  point  in  its  length  which  is  the  node  of  any  of  its  so-called 
harmonics,  those  simple  vibrational  forms  of  the  string  which  have  a  node  in  that 
point  are  not  contained  in  the  compound  vibrational  form.  Hence,  if  we  attack 
the  string  at  its  middle  point,  all  the  simple  vibrations  due  to  the  evenly  numbered 
partials,  each  of  which  has  a  note  at  that  point,  will  be  absent.  This  gives  the 
sound  of  the  string  a  peculiarly  hollow  or  nasal  twang.  If  Ave  excite  the  string  at 
1  of  its  length,  the  vibrations  corresponding  to  the  third,  sixth,  and  ninth  partials 
will  be  absent ;  if  at  },  then  those  corresponding  to  the  fourth,  eighth,  and  twelfth 
partials  will  fail ;  and  so  on.f 

This  result  of  mathematical  theory  is  confirmed,  in  the  first  place,  by  analys- 
ing the  compound  tone  of  the  string  by  sympathetic  resonance,  either  by  the 
f  resonators  or  by  other  strings.  The  experiments  may  be  easily  made  on  the 
pianoforte.  Press  down  the  digitals  for  the  notes  c  and  c,  without  allowing  the 
hammer  to  strike,  so  as  merely  to  free  them  from  their  dampers,  and  then  pluck 
the  string  c  with  the  nail  till  it  sounds.  On  damping  the  c  string  the  higher  c 
will  echo  the  sound,  except  in  the  particular  case  when  the  c  string  has  been 
plucked  exactly  at  its  middle  point,  which  is  the  point  where  it  would  have  to  be 
touched  in  order  to  give  its  first  harmonic  when  struck  by  the  hammer. 

If  yve  touch  the  c  string  at  i  or  |  its  length,  and  strike  it  with  the  luunmer, 
we  obtain  the  harmonic  g' ;  and  if  the  damper  of  the  g  is  raised,  this  string  echoes 
the  sound.  But  if  we  pluck  the  c  string  with  the  nail,  at  either  ^  or  |  its  length, 
g'  is  not  echoed,  as  it  will  be  if  the  c  string  is  plucked  at  any  other  spot. 

In  the  same  way  observations  with  the  resonators  show  that  when  the  r  string 

is  plucked  at  its  middle  the  Octave  c    is  missing,  and  when  at  i  or  |  its  length  the 

Twelfth  g    is  absent.     The  analysis   of  the  sound  of  a  string  by  the  sympathetic 

H  resonance  of  strings   or    resonators,    consequently  fully  confirms    Thomas  Young's 

law. 

But  for  the  vibration  of  strings  we  have  a  more  direct  means  of  analysis  than 
that  furnished  by  sympathetic  resonance.  If  we,  namely,  touch  a  vibrating  string 
gently  for  a  moment  with  the  finger  or  a  camel's-hair  pencil,  we  damp  all  those 
simple  vibrations  which  have  no  node  at  the  point  touched.  Those  vibrations, 
however,  which  have  a  node  there  are  not  damped,  and  hence  will  continue  to 
sound  without  the  others.  Consequently,  if  a  string  has  been  made  to  speak  in 
any  way  whatever,  and  we  wish  to  know  whether  there  exists  among  its  simple 
vibrations  one  corresponding  to  the  Twelfth  of  the  prime  tone,  we  need  only  touch 
one  of  the  nodes  of  this  vibrational  form  at  ^  or  |  the  length  of  the  string,  in 
order  to  reduce  to  silence  all  simple  tones  which  have  no  such  node,  and  leave  the 
Twelfth  sounding,   if  it  were   there.     If  neither  it,  nor  any  of  the   sixth,   ninth, 

*  London.     Philosophical  Transactions,  1800,  vol.  i.  p.  137. 
t  See  Appendix  III. 


CHAP.  IV.  PROOF  OF  OHM'S  LAAV.  53 

twelfth,   i\:c.,   of  the  partial   tones  were  present,   giving   corresponding   harmonics, 
the  string  will  be  reduced  to  absolute  silence  by  this  contact  of  the  finger. 

Press  down  one  of  the  digitals  of  a  piano,  in  order  to  free  a  string  from  its 
damper.  Pluck  the  string  at  its  middle  point,  and  immediately  touch  it  there. 
The  string  will  be  completely  silenced,  showing  that  plucking  it  in  its  middle 
excited  none  of  the  CA^enly  numbered  partials  of  its  compound  tone.  Pluck  it  at  }^  or  'f. 
its  length,  and  immediately  touch  it  in  the  same  place ;  the  string  will  be  silent, 
])roving  the  absence  of  the  third  partial  tone.  Pluck  the  string  anywhere  else 
than  in  the  points  named,  and  the  second  partial  will  be  heard  when  the  middle  is 
touched,  the  third  when  the  string  is  touched  at  i  or  ^  of  its  length. 

The  agreement  of  this  kind  of  proof  with  the  results  from  sympathetic  reso- 
nance, is  well  adapted  for  the  experimental  establishment  of  the  proposition  based 
in  the  last  chapter  solely  upon  the  results  of  mathematical  theory,  namely,  that 
sympathetic  vibration  occurs  or  not,  according  as  the  corresponding  simple^ 
vibrations  are  or  are  not  contained  in  the  compound  motion.  In  the  last  described 
method  of  analysing  the  tone  of  a  string,  we  are  quite  independent  of  the  theory 
of  sympathetic  vibration,  and  the  simple  vibrations  of  strings  are  exactl}"  charac- 
terised and  recognisable  by  their  nodes.  If  the  compound  tones  admitted  of  being 
analvsed  hj  sympathetic  resonance  according  to  any  other  vibrational  forms  except 
those  of  simple  vibration,  this  agreement  could  not  exist. 

If,  after  having  thus  experimentally  proved  the  correctness  of  Thomas  Young's 
law,  we  try  to  analyse  the  tones  of  strings  by  the  unassisted  ear,  we  shall  continue 
to  find  complete  agreement.*  If  we  pluck  or  strike  a  string  in  one  of  its  nodes, 
all  those  upper  partial  tones  of  the  compound  tone  of  the  string  to  which  the  node 
belongs,  disappear  for  the  ear  also,  bat  they  are  heard  if  the  string  is  plucked  at 
any  other  place.  Thus,  if  the  string  r  be  plucked  at  ^  its  length,  the  partial  tone 
f/'  cannot  be  heard,  but  if  the  string  be  plucked  at  only  a  little  distance  from  this 
point  the  partial  tone  (/'  is  distinctly  audible.  Hence  the  ear  analyses  the  sovmd  ^ 
of  a  string  into  precisely  the  same  constituents  as  are  found  by  sympathetic  reso- 
nance, that  is,  into  simple  tones,  according  to  Ohm's  definition  of  this  conception. 
These  experiments  are  also  well  adapted  to  show  that  it  is  no  mere  play  of  imagina- 
tion when  we  hear  upper  partial  tones,  as  some  people  believe  on  hearing  them  for 
the  first  time,  for  those  tones  are  not  heard  when  they  do  not  exist. 

The  following  modification  of  this  process  is  also  very  well  adapted  to  make 
the  upper  partial  tones  of  strings  audible.  First,  strike  alternately  in  rhythmical 
sequence,  the  third  and  fourth  partial  tone  of  the  string  alone,  by  damping  it  in  the 
corresponding  nodes,  and  request  the  listener  to  observe  the  simple  melody  thus 
produced.  Then  strike  the  luidamped  string  alternately  and  in  the  same  rhythmical 
sequence,  in  these  nodes,  and  thus  reproduce  the  same  melody  in  the  upper  partials, 
which  the  listener  will  then  easily  recognise.  Of  course,  in  order  to  hear  the 
third  partial,  we  must  strike  the  string  in  the  node  of  the  fourth,  and  conversely. 

The  compound  tone  of  a  plucked  string  is  also  a  remarkably  striking  example  H 
of  the  power  of  the  ear  to  analyse  into  a  long  series  of  partial  tones,  a  motion 
which  the  eye  and  the  imagination  are  able  to  conceive  in  a  much  simpler  manner. 
A  string,  which  is  pulled  aside  by  a  sharp  point,  or  the  finger  nail,  assumes  the 
form,  fig.  18,  A  (p.  54a),  before  it  is  released.  It  then  passes  through  the  series  of 
forms,  fig.  18,  B,  C,  D,  E,  F,  till  it  reaches  G,  which  is  the  inversion  of  A,  and 
then  returns,  through  the  same,  to  A  again.  Hence  it  alternates  between  the  forms 
A  and  G.  All  these  forms,  as  is  clear,  are  composed  of  three  straight  lines,  and 
on  expressing  the  velocity  of  the  individual  points  of  the  strings  by  vibrational 
curves,  these  would  have  the  same  form.  Now  the  string  scarcely  imparts  any 
})erceptible  portion  of  its  own  motion  directly  to  the  air.  Scarcely  any  audible 
tone  results  when  both  ends  of  a  string  are  fastened  to  immovable  supports,  as 
metal  bridges,  which  are  again  fastened  to  the   walls  of  a  room.     The  sound  of 

*  See  Brandt  in  PoggendorfE's  Annalcn  der  Fhijsik,  vol.  cxii.  p.  324,  where  this  fact  is 
proved. 

,2_ 


V 


<^i' 


o4 


PROOF  OF  OHM'S   LAW. 


tlie  string  reaches  the  air  through  that  one  of  its  extremities  which  rests  upon 
a  bridge  standing  on  an  elastic  sounding  board.  Hence  the  sound  of  the  .string 
essentially  depends  on  the  motion  of  this 
extremity,  through  the  pressure  which  it 
exerts  on  the  sounding  board.  The  magni- 
tude of  this  pressure,  as  it  alters  periodically 
with  the  time,  is  shown  in  fig.  19,  where 
the  height  of  the  line  h  h  corresponds  to 
the  amount  of  pressure  exerted  on  the  bridge 
by  that  extremity  of  the  string  when  the 
string  is  at  rest.  Along  h  h  suppose 
lengths  to  be  set  off  corresponding  to  con- 
secutive   intervals    of    time,    the    vertical 

^  heights  of  the  broken  line  above  or  below 
h  h  represent  the  corresponding  augmenta- 
tions or  diminutions  of  pressure  at  those 
times.  The  pressure  of  the  string  on  the 
sounding  board  consequently  alternates,  as 
the  figure  shows,  between  a  higher  and  a 
lower  value.  For  some  time  the  greater 
pressure  remains  unaltered  ;  then  the  lower 
suddenly  ensues,  and  likewise  remains  for  a 
time  unaltered.  The  letters  a  to  g  in  fig.  19 
correspond  to  the  times  at  which  the  string 
assumes  the  forms  A  to  G  in  fig.  18.  It  is  this  alteration  between  a  greater  and 
a  smaller  pressure  which  produces  the  sound  in  the  air.  We  cannot  but  feel 
astonished  that  a  motion  produced  by  means  so  simple  and  so  easy  to  comprehend, 

^  should  be  analysed  by  the  ear  into  such  a  complicated  sum  of  simple  tones.  For 
the  eye  and  the  understanding  the  action  of  the  string  on  the  sounding  board  can 
be  figured  with  extreme  simplicity.  What  has  the  simple  broken  line  of  fig.  19 
to   do    with    wave-curves,    which,    in    the    course    of   one    of    their   periods,    show 


h-|--l- 


de     fgfpde     ha 


1) 


3,  4,  5,  up  to  16,  and  more,  crests  and  troughs?  This  is  one  of  the  most  striking 
examples  of  the  different  ways  in  which  eye  and  ear  comprehend  a  periodic 
motion. 

There  is  no  sonorous  body  whose  motions  imder  varied  conditions  can  be  so 
^completely  calculated  theoretically  and  contrasted  with  observation  as  a  string. 
The  following  are  examples  in  which  theory  can  be  compared  with  analysis  by 
ear : — 

I  have  discovered  a  means  of  exciting  simple  pendular  vibrations  in  the  air.  A 
tuning-fork  when  struck  gives  no  harmonic  upper  partial  tones,  or,  at  most,  traces 
of  them  when  it  is  brought  into  such  excessively  strong  vibration  that  it  no  longer 
exactly  follows  the  law  of  the  pendulum.*  On  the  other  hand,  tuning  forks  have 
some  very  high   inharmonic  secondary  tones,  which   produce  that   peculiar  shaq) 

*  [On  all  ordinary  tuning-forks  between  a       pitch  numbers.     But  the  prime  can  always  be 


and  d"  in  pitch,  I  have  been  able  to  hear  the 
second  partial  or  Octave  of  the  prime.  In 
some  low  forks  this  Octave  is  so  powerful  that 
on  pressing  the  liaudle  of  the  fork  against  the 
table,  the  prime  quite  disappears  and  the 
Octave  only  is  heard,  and  this  has  often 
proved  a  source  of  embarrassment  in  tuning 
the  forks,  or  in  counting  beats  to  determine 


heard  when  the  fork  is  held  to  the  ear  or  over 
a  properly  tuned  resonance  jar,  as  described  in 
this  paragraph.  I  tune  such  jars  by  pouring 
water  in  or  out  until  the  resonance  is  strongest, 
and  then  I  register  the  height  of  the  water 
and  pitch  of  the  fork  for  future  use  on  a  slip 
of  paper  gummed  to  the  side  of  the  jar.  I 
have  found  that  it  is  not  at  all  necessary  to 


CHAP.  IV.  PROOF  OF  OHM'S  LAW.  55 

tinkling  of  the  fork  at  the  moment  of  being  struck,  and  generally  become  rapidly 
inaudible.  If  the  tuning  fork  is  held  in  the  fingers,  it  imparts  very  little  of  its 
tone  to  the  air,  and  cannot  be  heard  unless  it  is  held  close  to  the  ear.  Instead  of 
holding  it  in  the  fingers,  we  may  screw  it  into  a  thick  board,  on  the  inider  side  of 
which  some  pieces  of  india-rubber  tubing  have  been  fastened.  When  this  is  laid 
upon  a  table,  the  india-rubber  tubes  on  which  it  is  supported  convey  no  sound  to 
the  table,  and  the  tone  of  the  tuning-fork  is  so  weak  that  it  may  be  considered  in- 
audible. Now  if  the  prongs  of  the  fork  be  brought  near  a  resonance  chamber*  of 
a  bottle-form  of  such  a  size  and  shape  that,  when  we  blow  over  its  mouth,  the  air 
it  contains  gives  a  tone  of  the  same  pitch  as  the  fork's,  the  air  within  this  chamber 
vibrates  sympathetically,  and  the  tone  of  the  fork  is  thus  conducted  with  great 
strength  to  the  outer  air.  Now  the  higher  secondary  tones  of  such  resonance 
chambers  are  also  inharmonic  to  the  prime  tone,  and  in  general  the  secondary 
tones  of  the  chambers  correspond  neither  with  the  harmonic  nor  the  inharmonic  H 
secondary  tones  of  the  forks  ;  this  can  be  determined  in  each  particular  case  by 
producing  the  secondary  tones  of  the  bottle  by  stronger  blowing,  and  discovering 
those  of  the  forks  with  the  help  of  strings  set  into  sympathetic  vibration,  as  will 
be  presently  described.  If,  then,  only  one  of  the  tones  of  the  fork,  namely,  the 
prime  tone,  corresponds  Avith  one  of  the  tones  of  the  chamber,  this  alone  will  be 
reinforced  by  sympathetic  vibration,  and  this  alone  will  be  communicated  to  the 
external  air,  and  thus  conducted  to  the  observer's  ear.  The  examination  of  the 
motion  of  the  air  by  resonators  shows  that  in  this  case,  provided  the  tuning-fork  be 
not  set  into  too  violent  motion,  no  tone  but  the  prime  is  present,  and  in  such  case 
the  unassisted  ear  hears  only  a  single  simple  tone,  namely,  the  common  prime  of  the 
tuning-fork  and  of  the  chamber,  without  any  accompanying  upper  partial  tones. 

The  tone  of  a  tuning-fork  can  also  be  purified  from  secondary  tones  by  placing 
its  handle  upon  a  string  and  moving  it  so  near  to  the  bridge  that  one  of  the  proper 
tones  of  the  section  of  string  lying  between  the  fork  and  the  bridge  is  the  same  as  ^ 
that  of  the  tuning-fork.  The  string  then  begins  to  vibrate  strongly,  and  conducts 
the  tone  of  the  tuning-fork  with  great  power  to  the  sounding  board  and  surround- 
ing air,  whereas  the  tone  is  scarcely,  if  at  all,  heard  as  long  as  the  above-named 
section  is  not  in  unison  with  the  tone  of  the  fork.  In  this  way  it  is  easy  to  find 
the  lengths  of  string  which  correspond  to  the  prime  and  upper  partial  tones  of  the 
fork,  and  accurately  determine  the  pitch  of  the  latter.  If  this  experiment  is  con- 
ducted with  ordinary  strings  which  are  uniform  throughout  their  length,  we  shield 
the  ear  from  the  inharmonic  secondary  tones  of  the  fork,  but  not  from  the  harmonic 
upper  partials,  which  are  sometimes  faintly  present  when  the  fork  is  made  to 
vibrate  strongly.  Hence  to  conduct  this  experiment  in  such  a  way  as  to  create 
purely  pendular  vibrations  of  the  air,  it  is  best  to  weight  one  point  of  the  string,  if 
only  so  much  as  by  letting  a  drop  of  melting  sealing-wax  fall  upon  it.  This  causes 
the  upper  proper  tones  of  the  string  itself  to  be  inharmonic  to  the  prime  tone,  and 
hence  there  is  a  distinct  interval  between  the  points  where  the  fork  must  be  placed  U 
to  bring  out  the  prime  tone  and  its  audible  Octave,  if  it  exists.  ' 

In  most  other  cases  the  mathematical  analysis  of  the  motions  of  soimd  is  not 
nearly  far  enough  advanced  to  determine  with  certainty  what  upper  partials  will 
be  present  and  what  intensity  they  will  possess.  In  circular  plates  and  stretched 
membranes  which  are  struck,  it  is  theoretically  possible  to  do  so,  but  their  inhar- 

put  the  fork  into  excessively  strong  vibration  of  Chap.  VII.,  and  Prof.  Preyer's  in  App.  XX. 

in  order  to  make  the  Octave  sensible.     Thus,  sect.  L.   art.  4,  <%     The  conditions  according 

taking  a  fork  of  232  and  another  of  468  vibra-  to  Koenig  that  tuning-forks  should  have  no 

tions,  after  striking  them  both,  and  letting  the  upper  partials  are  given  in  App.  XX.  sect.  L. 

deeper  fork  spend  most  of  its  energy  until  I  art.  2,  «. —  Translator.] 

could  not  see  the  vibrations  with  the  eye  at  all,  *  Either  a  bottle  of  a  proper  size,  which 
the  beats  were  heard  distinctly,  when  I  pressed  can  readily  be  more  accurately  tuned  by  pour- 
both  on  to  a  table,  and  continued  to  be  heard  ing  oil  or  water  into  it,  or  a  tube  of  pasteboard 
even  after  the  forks  themselves  were  separately  quite  closed  at  one  end,  and  having  a  small 
inaudible.  See  also  Prof.  Helmholtz's  experi-  round  opening  at  the  other.  See  the  proper 
ments  on  a  fork  of  64  vibrations  at  the  close  sizes  of  such  resonance  chambers  in  App.  IV. 


56 


PROOF  OF  OHM'S  LAW. 


monic  secondary  tones  are  so  numerous  and  so  nearly  of  the  same  pitch  that  most 
observers  would  probably  fail  to  separate  them  satisfactorily.  On  elastic  rods,  how- 
ever, the  secondary  tones  are  very  distant  from  each  other,  and  are  inharmonic,  so 
that  they  can  be  readily  distinguished  from  each  other  by  the  ear.  The  following 
are  the  proper  tones  of  a  rod  which  is  free  at  both  ends ;  the  A'ibrational  number 
of  the  prime  tone  taken  to  be  c,  is  reckoned  as  1 : — 


Pitch  Number 

Cents  * 

Notation 

Prime  tone 

Second  proper  tone        ..... 

Third  proper  tone 

Fourth  proper  tone 

1-0000 
2-7570 
5-4041 
13-3444 

0 

1200  +  556 
2400  +  521 
3600  +  886 

/      +0-2 
f"     +0-1 
V"    -0-1 

The  notation  is  adapted  to  the  equal  temperament,  and  the  appended  fractious 
H  are  parts  of  the  interval  of  a  complete  tone. 

Where  we  are  unable  to  execute  the  theoretical  analysis  of  the  motion,  we  can, 
at  any  rate,  by  means  of  resonators  and  other  sympathetically  vibrating  bodies, 
analyse  any  individual  musical  tone  that  is  produced,  and  then  compare  this 
analysis,  which  is  determined  by  the  laws  of  sympathetic  vibration,  with  that 
effected  by  the  unassisted  ear.  The  latter  is  naturally  much  less  sensitive  than 
one  armed  with  a  resonator ;  so  that  it  is  frequently  impossible  for  the  unarmed 
ear  to  recognise  amongst  a  number  of  other  stronger  simple  tones  those  which  the 
resonator  itself  can  only  faintly  indicate.  On  the  other  hand,  so  far  as  my  ex- 
perience goes,  there  is  complete  agreement  to  this  extent :  the  ear  recognises  with- 
out resonators  the  simple  tones  which  the  resonators  greatly  reinforce,  and  perceives 
no  upper  partial  tone  which  the  resonator  does  not  indicate.  To  verify  this  con- 
clusion, I  performed  numerous  experiments,  both  with  the  human  voice  and  the 
harmonium,  and  they  all  confirmed  it.+ 
^  By  the  above  experiments  the  proposition  enunciated  and  defended  by  G.  S. 
Ohm  must  be  regarded  as  proved,  viz.  that  the  human  ear  perceives  2)endidar  vibra- 
tions alone  as  simple  tones,  and  resolves  all  other  periodic  motions  of  the  air  into  a 
series  of  pendular  viM'ations,  hearing  the  series  of  simple  tones  which  correspond,  ivith 
these  simple  vibrations. 

Calling,  then,  as  already  defined  (in  pp.  2.3,  24  and  note),  the  sensation  excited 
in  the  ear  by  any  periodical  motion  of  the  air  a  musical  tone,  and  the  sensation 
excited  by  a  simple  pendular  vibration  a  simp)le  tone,  the  rule  asserts  that  the 
serisation  of  a  mtisical  tone  is  compounded  out  of  the  sensations  of  several  simple 
tones.  In  particular,  we  shall  henceforth  call  the  sound  produced  by  a  single 
sonorous  body  its  (simple  or  compound)  tone,  and  the  sound  produced  by  several 
musical  instruments  acting  at  the  same  time  a  composite  tone,  consisting  generally 
of  several  (simple  or   compound)   tones.     If,  then,  a   single    note  is  sounded  on  a 

reed  tones,  by  the  beats  (Chap.  VIII.)  that  their 
upper  partials  made  with  the  primes  of  a  set  of 
Scheibler's  tuning-forks.  The  correctness  of 
the  process  was  proved  by  the  fact  that  the 
results  obtained  from  different  partials  of  the 
same  reed  tone,  which  were  made  to  beat  with 
different  forks,  gave  the  same  pitch  numbers 
for  the  primes,  within  one  or  two  hundredths  of 
a  vibration  in  a  second.  I  not  only  employed 
such  low  partials  as  3,  4,  5  for  one  tone,  and 
4,  5,  6  for  others,  but  I  determine  the  pitch 
number  31-47,  by  partials  7,  8,  9,  10,  11,  12, 
13,  and  the  pitch  number  15-94  by  partials  25 
and  27.  The  objective  reality  of  these  ex- 
tremely high  upper  partials,  and  their  inde- 
pendence of  resonators  or  resonance  jars,  was 
therefore  conclusively  shown.  On  the  Har- 
monical  the  beats  of  the  16th  partial  of  C  66, 
with  c'",  when  slightly  flattened  bypressing  the 
note  lightly  down,  are  very  clear.-  -Translator.] 


H  *  [For  cents  see  note  p.  i\d.  As  a  Tone  is 
200  ct.,  0-1  Tone  =  20ct.,  these  would  give  for 
the  Author's  notation  /'  +  40  ct.,  /"  +  20  ct.,  «'" 
- 10  ct.,  whereas  the  column  of  cents  shows 
that  they  are  more  accurately  /'  +  56  ct. ,  /"  + 
21  ct.,  a'"  -  14  ct.  For  convenience,  the  cents 
for  Octaves  are  separated,  thus  1200  +  556  in 
place  of  1756,  but  this  separation  is  quite 
unnecessary.  The  cents  again  show  the  inter- 
vals of  the  inharmonic  partial  tones  without 
any  assumption  as  to  the  value  of  the  prime. 
By  a  misprint  in  all  the  German  editions, 
followed  in  the  first  English  edition,  the  second 
proper  tone  was  made  /"'  -  0-2  in  place  of  /'  + 
0-2.— Translator.] 

t  [In  my  '  Notes  of  Observations  on  Musi- 
cal Beats,'  Proceedings  of  the  Royal  Socirfi/, 
May,  1880,  vol.  xxx.  p.  531,  largely  cited  in 
App.  XX.  sect.  B.  No.  7,  I  showed  that  I  was 
able  to  determine  the  pitch  numbers  of  deep 


CHAP.  IV.  DIFFICULTIES  IX  OBSERVING  PARTIALS.  57 

musical  instruinent,  as  a  violin,  tninipot,  organ,  or  by  a  singing  voice,  it  must  bo 
called  in  exact  language  a  tone  of  the  instrument  in  ([uestion.  This  is  also  the 
ordinary  language,  but  it  did  not  then  imply  that  the  tone  migiit  be  mnipound. 
When  the  tone  is,  as  usual,  a  compound  tone,  it  will  be  distinguished  by  this  term, 
or  the  abridgment,  a  compound ;  while  tone  is  a  general  term  which  includes  both 
simple  and  compound  tones.*  The  prime  tone  is  generally  louder  than  any  of  the 
upper  partial  tones,  and  hence  it  alone  generally  determines  the  ^>«'fcy^  of  the  com- 
pound. The  tone  produced  by  any  sonorous  body  reduces  to  a  migle  simple  tone 
in  very  few  cases  indeed,  as  the  tone  of  tuning-forks  imparted  to  the  air  by  reso- 
nance chambers  in  the  manner  already  described.  The  tones  of  wide-stopped 
organ  pipes  when  gently  blown  are  almost  free  from  upper  partials,  and  are  accom- 
panied only  by  a  rush  of  wind. 

It  is  well  known  that  this  union  of  several  simple  tones  into  one  compound 
tone,  which  is  naturally  effected  in  the  tones  produced  by  most  musical  instruments,  ^ 
is  artiricially  imitated  on  the  organ  by  peculiar  mechanical  contrivances.  The 
tones  of  organ  pipes  are  comparatively  poor  in  upper  partials.  When  it  is  desirable 
to  use  a  stop  of  incisive  penetrating  quality  of  tone  and  great  power,  the  wide  pipes 
{principal  re;/ister  and  weitgedacJd  f)  are  not  sufficient ;  their  tone  is  too  soft,  too 
defective  in  upper  partials  ;  and  the  narrow-pipes  {geigen-register  and  qidntaten  %) 
are  also  unsuitable,  because,  although  more  incisive,  their  tone  is  weak.  For  such 
occasions,  then,  as  in  accompanying  congregational  singing,  recourse  is  had  to  the 
compound  stops.^  In  these  stops  every  key  is  connected  with  a  larger  or  smaller 
series  of  pipes,  which  it  opens  simultaneously,  and  which  give  the  prime  tone  and 
a  certain  number  of  the  lower  upper  partials  of  the  compound  tone  of  the  note  in 
question.  It  is  very  usual  to  connect  the  upper  Octave  with  the  prime  tone,  and 
after  that  the  Twelfth.  The  more  complex  compounds  (cor/t^^i;  i)  give  the  first  six 
partial  tones,  that  is,  in  addition  to  the  two  Octaves  of  the  prime  tone  and  its 
Twelfth,  the  higher  major  Third,  and  the  Octave  of  the  Twelfth.  This  is  as  much  H 
of  the  series  of  upper  partials  as  belongs  to  the  tones  of  a  major  chord.  But 
to  prevent  these  compound  stops  from  being  insupportably  noisy,  it  is  necessary 
to  reinforce  the  deeper  tones  of  each  note  by  other  rows  of  pipes,  for  in  all  natural 
tones  which  are  suited  for  musical  purposes  the  higher  partials  decrease  in  force  as 
they  rise  in  pitch.  This  has  to  be  regarded  in  their  imitation  by  compound  stops. 
These  compound  stops  were  a  monster  in  the  path  of  the  old  musical  theory,  which 
was  acquainted  only  with  the  prime  tones  of  compounds ;  but  the  practice  of 
organ-builders  and  organists  necessitated  their  retention,  and  when  they  are 
suitably  arranged  and  properly  applied,  they  form  a  very  effective  musical  apparatus. 

*  [Here,  again,  as  on  pp.  23,  24,  I  have,  in  toned  diapason,  eight  feet.'     Hopkins,  Organ, 

the  translation,  been   necessarily  obliged  to  p.  445.     '  A  manual  stop  of  eight  feet,  produ- 

deviate  slightly  from  the  original.     K/.aiuj,  as  cing  a  pungent   tone  very  lilce    that   of   the 

here  defined,  embraces   Ton  as  a   particular  Gamba,  except  that  the  pipes,  being  of  larger 

case.      I  use  tunc  for  the  general  term,  and  scale,  speak  quicker  and  produce  a  fuller  tone. 

wmpoiuul   tone   and  simple   tone   for   the   two  Examples   of   the   stop   exist    at    Doncaster,  If 

particular  cases.  Thus,  as  ijresently  mentioned  the   Temple   Church,   and   in   the   Exchange 

in  the  text,  the  tone  produced  by  a  tuning-fork  Organ  at  Northampton.'     Ibid.  p.  138.     For 

held  over  a  proper  resonance  chamber  we  know,  quintaten,  see  supra,  p.  S3d,  note. — Translator.^ 
on  analysis,  to  be  simple,  but  before  analysis  it  §  [As  described  in  Hopkins,  Organ,  p.  142, 

is  to  us  only  a  (musical)  tone  like  any  other,  these  are  the  scsquialtera  '  of  five,  four,  three, 

and  hence  in  this  case  the  Author's  Klamj  or   two   ranks    of    open    metal   pipes,   tuned 

becomes  the  Author's  Ton.     I  believe  that  the  in  Thirds,  Fifths,  and  Octaves  to  the  Diapa- 

language  used  in  my  translation  is  best  adapted  son'.     The  mixture,  consisting  of  five  to  two 

for  the  constant  accurate  distinction  between  ranks  of  open  metal  pipes  smaller  than  the 

compound  and  simple  tones  by  English  readers,  last,  is  in  England  the  second,  in  Germany  the 

as  I  leave  nothing  which  runs  counter  to  old  first,  compound  stop  (p.  143).    The  Furniture  of 

habits,  and  by  the  use  of  the  words  simple  and  five  to  two  sets  of  small  open  pipes,  is  variable, 

compound,  constantly  recall  attention  to  this  (1)  The  Cornet,  vwnnted,  has  five  ranks  of  very 

newly  discovered  and  extremely  important  rela-  large  and  loudly  voiced  pipes,  (2)  the  echo  is 

tion. — Transltdor.]  similar,  but  light  and  delicate,  and  is  enclosed 

t  [Pr//*rv>(/— double  open  diapason.    Gross-  in  a  box.     In  German  organs  the  cornet  is  also 

gedackt—dou.h\e  stopped  diapason.     Hopkins,  a  pedal  reed  stop  of  four  and  two  feet  ((6/d). — 

Organ,  p.  444-5. —  Translator.]  Translator.] 

t  ['  Oeigcn     Principal  —  violin     or     crisp- 


58  DIFFICULTIES  IN  OBSERVING  PARTIALS.  part  i. 

The  nature  of  the  case  at  the  same  time  fully  justifies  their  use.  The  musician  is 
bound  to  regard  the  tones  of  all  musical  instruments  as  compomided  in  the  same 
way  as  the  compound  stops  of  organs,  and  the  important  part  this  method  of  com- 
position plays  in  the  construction  of  musical  scales  and  chords  will  be  made  evident 
in  subsequent  chapters. 

We  have  thus  been  led  to  an  appreciation  of  upper  partial  tones,  which  diflers 
considerably  from  that  previously  entertained  by  musicians,  and  even  physicists, 
and  must  therefore  be  prei)ared  to  meet  the  opposition  which  will  be  raised.  The 
upper  partial  tones  were  indeed  known,  but  almost  only  in  such  compound  tones  as 
those  of  strings,  where  there  was  a  favourable  opportunity  for  observing  them ; 
but  they  appear  in  previous  physical  and  musical  works  as  an  isolated  accidental 
phenomenon  of  small  intensity,  a  kind  of  curiosity,  which  was  certainly  occasion- 
ally adduced,  in  order  to  give  some  support  to  the  opinion  that  nature  had  pre- 
H  figured  the  construction  of  our  major  chord,  but  which  on  the  whole  remained 
almost  entirely  disregarded.  In  opposition  to  this  we  have  to  assert,  and  we  shall 
prove  the  assertion  in  the  next  chapter,  that  upper  partial  tones  are,  with  a  few 
exceptions  already  named,  a  general  constituent  of  all  musical  tones,  and  that  a 
certain  stock  of  upper  partials  is  an  essential  condition  for  a  good  musical  quality 
of  tone.  Finally,  these  upper  partials  have  been  erroneously  considered  as  weak, 
because  they  are  difficult  to  observe,  while,  in  point  of  fact,  for  some  of  the  best 
musical  qualities  of  tone,  the  loudness  of  the  first  upper  partials  is  not  far  inferior 
to  that  of  the  prime  tone  itself. 

There  is  no  difficulty  in  verifying  this  last  fact  by  experiments  on  the  tones  of 
strings.  Strike  the  string  of  a  piano  or  monochord,  and  immediately  touch  one  of 
its  nodes  for  an  instant  with  the  finger ;  the  constituent  partial  tones  having  this 
node  will  remain  with  unaltered  loudness,  and  the  rest  will  disappear.  We  might 
also  touch  the  node  in  the  same  way  at  the  instant  of  striking,  and  thus  obtain  the 
f  corresponding  constituent  partial  tones  from;,  the  first,  in  place  of  the  complete 
compound  tone  of  the  note.  In  both  ways  we  can  readily  convince  ourselves  that  the 
first  upper  partials,  as  the  Octave  and  Twelfth,  are  by  no  means  weak  and  difficult 
to  hear,  but  have  a  very  appreciable  strength.  In  some  cases  we  are  able  to  assign 
numerical  values  for  the  intensity  of  the  upper  partial  tones,  as  will  be  shown  in 
the  next  chapter.  For  tones  not  produced  on  strings  this  a  posteriori  proof  is  not 
so  easy  to  conduct,  because  we  are  not  able  to  make  the  upper  partials  speak 
separately.  But  even  then  by  means  of  the  resonator  we  can  apjjreciate  the  in- 
tensity of  these  upper  partials  by  producing  the  corresponding  note  on  the  same 
or  some  other  instrument  until  its  loudness,  when  heard  through  the  resonator, 
agrees  with  that  of  the  former. 

The  difficulty  we  experience  in  hearing  upper  partial  tones  is  no  reason  for 
considering  them  to  be  weak  ;  for  this  difficulty  does  not  depend  on  their  intensity, 
but  upon  entirely  diffi^rent  circumstances,  which  could  not  be  properly  estimated 
51  until  the  advances  recently  made  in  the  physiology  of  the  senses.  On  this  diffi- 
culty of  observing  the  upper  partial  tones  have  been  founded  the  objections  which 
A.  Seebeck*  has  advanced  against  Ohm's  law  of  the  decomposition  of  a  musical 
tone ;  and  perhaps  many  of  my  readers  who  are  unacquainted  with  the  physiology 
of  the  other  senses,  particularly  with  that  of  the  eye,  might  be  inclined  to  adopt 
Seebeck's  opinions.  I  am  therefore  obliged  to  enter  into  some  details  concerning 
this  difference  of  opinion,  and  the  peculiarities  of  the  perceptions  of  our  senses, 
on  which  the  solution  of  the  difficulty  depends. 

Seebeck,  although  extremely  accomplished  in  acoustical  experiments  and 
observations,  was  not  always  able  to  recognise  upper  partial  tones,  where  Ohm's 
law  required  them  to  exist.  But  we  are  also  bound  to  add  that  he  did  not  apply 
the  methods  already  indicated  for  directing  the  attention  of  his  ear  to  the  upper 
partials  in  question.  In  other  cases  when  he  did  hear  the  theoretical  upper 
*  In  PoggendorfE's  Annaloi  der  Physik,  vol.  Ix.  p.  449,  vol.  Ixiii.  pp.  353  and  368.— Ohm, 
ibid.  vol.  lix.  p.  513,  and  vol.  Ixii.  p.  1. 


CHAP.  IV.  DIFFICULTIES  IN  OBSERVINO  PARTIALS.  59 

pavtials,  they  were  weaker  than  the  theory  required.     He  conchided  that  the  defi- 
nition of  a  simple  tone  as  given  by  Ohm  was  too  limited,  and  that  not  only  pcn- 
dular  vibrations,  but  other  vibrational  forms,  provided  they  were  not  too  widely 
separated  from  the  pendular,  were  capable  of  exciting  in  the  ear  the  sensation  of 
a  single  simple  tone,  which,   however,   had  a   variable  quality.     He   consequently 
asserted  that  when  a  musical  tone  was  compounded  of  several  simple  tones,  part 
of  the  intensity  of  the   u])per  constituent  tones  went   to  increase  the  intensity  of 
the  prime  tone,  with  which  it  fused,  and  that  at  most  a  small  remainder  excited  in 
the  ear  the  sensation  of  an  upper  partial  tone.     He  did  not  formulate  any  deter- 
minate law,  assigning  the  vibrational  forms  which  would    give    the  impression  of 
a  simple  and  those  which  would  give  the  impression    of  a  compound  tone.     The 
experiments  of  Seebeck,   on  which   he   founded   his   assertions,   need   not  be   here 
described  in  detail.     Their  object   was    only  to  produce   musical   tones  for  which 
either  the   intensity  of  the   simple  vibrations   corresponding  to  the  upper  partialsll 
could    be    theoretically    calculated,    or    in    which    these    upper   partials    could    be 
rendered  separately  audible.     For  the  latter  purpose  the  siren  was  used.     We  have 
just  described  how  the  same  object  can  be  attained  by  means  of  strings.     Seebeck 
shows  in  each  case  that  the  simple  vibrations  corresponding  to  the  upi)er  partials 
have  considerable  strength,  but  that  the  upper  partials  are  either  not  heard  at  all, 
or  heard  with  difficulty  in  the  compound  tone  itself.     This  fact  has  been  already 
mentioned  in  the  present  chapter.     It  may  be  perfectly  true  for  an  observer  who 
has  not  applied  the  proper  means   for  observing   upper  partials,  while  another,  or 
even  the  first  observer  himself  when  properly  assisted,  can  hear  them  perfectly  well.* 
Now   there    are    many   circumstances   which    assist  us   first   in   separating   the 
musical  tones  arising  from  different  sources,  and  secondly,  in  keeping  together  the 
partial  tones  of  each  se[)arate  source.     Thus  when  one  musical  tone  is  heard  for 
some  time  before  being  joined  by  the  second,  and  then  the  second  continues  after 
the  first  has  ceased,  the  separation  in  sound  is  facilitated  by  the  succession  of  time.  H 
We  have   already  heard  the   first   musical   tone  by  itself,  and  hence  know  inune- 
diately  what  we  have  to  deduct  from  the  compound  effect  for  the  effect  of  this  first 
tone.     Even  when  several  parts  proceed  in  the  same  rhythm  in  polyphonic  music, 
the   mode   in  which  the   tones  of  different  instruments  and  voices  commence,  the 
nature   of  their  increase  in  force,  the  certainty  with  which  they  are  held,  and  the 
manner  in  which  they  die  off,  are  generally  slightly  different  for  each.     Thus  the 
tones    of   a    pianoforte    commence    suddenly  with    a   blow,    and    are    consequently 
strongest  at   the  first  moment,  and  then  rapidly  decrease  in  power.     The  tones  of 
brass   instruments,  on  the  other  hand,  commence  sluggishly,  and  require  a  small 
but  sensible  time  to  develop  their  full  strength.     The  tones  of  bowed  instruments 
are   distinguished   by   their  extreme   mobility,   but  when  either  the  player  o    the 
instrument    is    not    unusually   perfect    they  are  interrupted  by  little,  vei-y  short, 
pauses,  producing  in  the  ear  the  sensation  of  scraping,  as  will  be  described  more 
in  detail  when  we  come  to  analyse  the  musical  tone  of  a  violin.     When,  then,  such  ^ 
instruments  are  sounded  together  there  are  generally  points  of  time  when  one  or 
the  other  is  predominant,  and  it  is  consequently  easily  distinguished  by  the  ear. 
Bat  besides  all  this,  in  good  part   music,  especial  care  is  taken  to  facilitate    the 
separation  of  the  parts  b}'  the  ear.     In  polyphonic  music  proper,  where  each  part 
has  its  own  distinct  melody,  a  principal  means  of  clearly  separating  the  progres- 
sion of  each  part  has  always  consisted  in  making  them  proceed  in  different  rhythms 
and  on  different  divisions  of  the  bars ;  or  where  this  could  not  be  done,  or  was  at 
any  rate  only  partly  possible,  as  in  four-part  chorales,  it  is  an  old  rule,  contrived 
for  this  purpose,  to  let  three  parts,  if  possible,  move  by  single  degrees  of  the  scale, 
and  let  the  fourth  leap  over  several.     The  small  amount  of  alteration  in  the  pitch 
makes  it  easier  for  the  listener  to  keep  the  identity  of  the  several  voices  distinctly 
in  mind. 

*  [Here  ivom  '  Upper  partial  tones,'  p.  94,  to  '  former  analysis,'  p.  100  of  the  1st  English 
edition  are  omitted,  in  accordance  with  the  4th  German  edition.  —  'I'ranslafor.] 


60 


FUSION  OF  PARTIALS  INTO  A  COMPOUND. 


All  these  helps  fail  in  the  resolution  of  musical  tones  into  their  constituent 
partials.  When  a  compound  tone  commences  to  sound,  all  its  partial  tones 
commence  with  the  same  comparative  strength  ;  when  it  swells,  all  of  them 
generally  swell  uniformly;  when  it  ceases,  all  cease  simultaneously.  Hence  no 
opportunity  is  generally  given  for  hearing  them  separately  and  independently.  In 
precisely  the  same  manner  as  the  naturally  connected  partial  tones  form  a  single 
source  of  sound,  the  partial  tones  in  a  compound  stop  on  the  organ  fuse  into  one,  as 
all  are  struck  with  the  same  digital,  and  all  move  in  the  same  melodic  progression 
as  their  prime  tone. 

Moreover,  the  tones  of  most  instruments  are  usually  accompanied  by  charac- 
teristic irregular  noises,  as  the  scratching  and  rubbing  of  the  violin  bow,  the  rush 
of  wind  in  flutes  and  organ  pipes,  the  grating  of  reeds,  &c.  These  noises,  with 
wdiich  we  are  already  familiar  as  characterising  the  instruments,  materially 
^  facilitate  our  power  of  distinguishing  them  in  a  composite  mass  of  sounds.  The 
partial  tones  in  a  compound  have,  of  course,  no  such  characteristic  marks. 

Hence  we  have  no  reason  to  be  surprised  that  the  resolution  of  a  compound 
tone  into  its  partials  is  not  quite  so  easy  for  the  ear  to  accomplish,  as  the  resolu- 
tion of  composite  masses  of  the  musical  sounds  of  many  instruments  into  their 
proximate  constituents,  and  that  even  a  trained  musical  ear  requires  the  applica- 
tion of  a  considerable  amount  of  attention  when  it  undertakes  the  former  problem. 

It  is  easy  to  see  that  the  auxiliary  circumstances  already  named  do  not  always 
sufhce  for  a  correct  separation  of  musical  tones.  In  uniformly  sustained  musical 
tones,  where  one  might  be  considered  as  an  upper  partial  of  another,  onr 
judgment  might  readily  make  default.  This  is  really  the  case.  G.  S.  Ohm 
proposed  a  very  instructive  experiment  to  show  this,  using  the  tones  of  a  violin. 
But  it  is  more  suitable  for  such  an  experiment  to  use  simple  tones,  as  those  of  a 
stopped  organ  pipe.  The  best  instrument,  however,  is  a  glass  bottle  of  the  form 
H  shown  in  fig.  20,  which  is  easily  procured  and 
prepared  for  the  experiment.  A  little  rod  c 
supports  a  guttapercha  tube  a  in  a  proper 
position.  The  end  of  the  tube,  which  is 
directed  towards  the  bottle,  is  softened  in  warm 
water  and  pressed  flat,  forming  a  narrow  chink, 
through  which  air  can  be  made  to  rush  over 
the  mouth  of  the  bottle.  When  the  tube  is 
fastened  by  an  india-rubber  pipe  to  the  nozzle 
of  a  bellows,  and  wind  is  driven  over  the  bottle, 
it  produces  a  hollow  obscure  sound,  like  the 
vowel  00  in  too,  which  is  freer  from  upper 
partial  tones  than  even  the  tone  of  a  stopped 
pipe,  and  is  only  accompanied  by  a  slight 
m  noise  of  wind.  I  find  that  it  is  easier  to  keep 
the  pitch  unaltered  in  this  instrument  wliile 
the  pressure  of  the  wind  is  slightly  changed, 
than  in  stopped  pipes.  We  deepen  the  tone  by  fr 
partially  shading  the  orifice  of  the  bottle  with  t_^ 
a  little  wooden  plate ;  and  we  sharpen  it  by 
pouring  in  oil  or  melted  wax.  We  are  thus  able  tcj  make  any  required  little 
alterations  in  pitch.  I  tuned  a  large  bottle  to  b'^  and  a  smaller  one  to  b'b  and 
united  them  with  the  same  bellows,  so  that  when  used  both  began  to  speak  at  the 
same  instant.  When  thus  united  they  gave  a  musical  tone  of  the  pitch  of  the 
deeper  />[?,  but  having  the  quality  of  tone  of  the  vowel  oa  in  toad,  instead  of  oo  in 
too.  When,  then,  I  compressed  first  one  of  the  india-rubber  tubes  and  then  the 
other,  so  as  to  produce  the  tones  alternately,  separately,  and  in  connection,  I  was 
at  last  able  to  hear  them  separately  when  sounded  together,  but  I  could  not 
continue  to  hear  them  separately  for  long,  for  the  upper  tone  gradually  fused  with 


I 


CHAP.  IV.  SEPARATION  OF  THE  PARTIALS.  lU 

the  lower.  Tliis  fusion  takes  place  even  when  the  upper  tone  is  somewhat  stronger 
than  the  lower.  The  alteration  in  the  quality  of  tone  which  takes  place  during 
this  fusion  is  characteristic.  On  producing  the  upper  tone  first  and  then  letting 
the  lower  sound  with  it,  I  found  that  I  at  first  continued  to  hear  the  upper  tone 
with  its  full  force,  and  the  under  tone  sounding  below  it  in  its  natural  quality  of 
oo  in  too.  But  by  degrees,  as  my  recollection  of  the  sound  of  the  isolated  upper 
tone  died  away,  it  seemed  to  become  more  and  more  indistinct  and  weak,  while 
the  lower  tone  appeared  to  become  stronger,  and  sounded  like  oa  in  toad.  This 
weakening  of  the  upper  and  strengthening  of  the  lower  tone  was  also  observed  by 
Ohm  on  the  violin.  As  Seebeck  remarks,  it  certainly  does  not  always  occur,  and 
probably  depends  on  the  liveliness  of  our  recollections  of  the  tones  as  heard 
separately,  and  the  greater  or  less  uniformity  in  the  simultaneous  production  of 
the  tones.  But  where  the  experiment  succeeds,  it  gives  the  best  proof  of  the 
essential  dependence  of  the  result  on  varying  activity  of  attention.  With  the  tones  H 
produced  by  bottles,  in  addition  to  the  reinforcement  of  the  lower  tone,  the  altera- 
tion in  its  quality  is  very  evident  and  is  characteristic  of  the  nature  of  the  process. 
This  alteration  is  less  striking  for  the  penetrating  tones  of  the  violin.* 

This  experiment  has  been  appealed  to  both  by  Ohm  and  by  Seebeck  as  ;i 
cDrroboratiou  of  their  different  opinions.  When  Ohm  stated  that  it  was  an 
'  illusion  of  the  ear '  to  apprehend  the  upper  partial  tones  wholly  or  partly  as  a 
reinforcement  of  the  prime  tone  (or  rather  of  the  compound  tone  whose  pitch  is 
determined  by  that  of  its  prime),  he  certainly  vised  a  somewhat  incorrect  expression, 
although  he  meant  what  was  correct,  and  Seebeck  was  justified  in  replying  that 
the  ear  was  the  sole  judge  of  auditory  sensations,  and  that  the  mode  in  which  it 
apprehended  tones  ought  not  to  be  called  an  '  illusion '.  However,  our  experiments 
just  described  show  that  the  judgment  of  the  ear  differs  according  to  the  liveliness 
of  its  recollection  of  the  separate  auditory  impressions  here  fused  into  one  whole, 
and  according  to  the  intensity  of  its  attention.  Hence  we  can  certainly  appeal  from  H 
the  sensations  of  an  ear  directed  without  assistance  to  external  objects,  whose 
interests  Seebeck  represents,  to  the  ear  which  is  attentively  observing  itself  and 
is  suitably  assisted  in  its  observation.  Such  an  ear  really  proceeds  according  to 
the  law  laid  down  by  Ohm. 

Another  experiment  should  be  adduced.  Raise  the  dampers  of  a  pianoforte  so 
that  all  the  strings  can  vibrate  freely,  then  sing  the  vowel  a  in  father,  art,  loudly 
to  any  note  of  the  piano,  directing  the  voice  to  the  sounding-board ;  the  sym- 
pathetic resonance  of  the  strings  distinctly  re-echoes  the  same  a.  On  singing  oe 
in  toe,  the  same  oe  is  re-echoed.  On  singing  a  in  fare,  this  a  is  re-echoed.  For  ee 
in  see  the  echo  is  not  quite  so  good.  The  experiment  does  not  succeed  so  well  if 
the  damper  is  removed  only  from  the  note  on  which  the  vowels  are  sung.  The 
vowel  character  of  the  echo  arises  from  the  re-echoing  of  those  upper  partial  tones 
which  characterise  the  vowels.  These,  however,  will  echo  better  and  more 
clearly  when  their  corresponding  higher  strings  are  free  and  can  vibrate  sym-^ 
pathetically.  In  this  case,  then,  in  the  last  resort,  the  musical  effect  of  the 
resonance  is  compounded  of  the  tones  of  several  strings,  and  several  separate 
partial  tones  combine  to  produce  a  musical  tone  of  a  peculiar  quality.  In  addition 
to  the  vowels  of  the  human  voice,  the  piano  will  also  quite  distinctly  imitate  the 
quality  of  tone  ])roduced  by  a  clarinet,  when  strongly  blown  on  to  the  sounding- 
board. 

Finally,  we  must  remark,  that  although  the  pitch  of  a  compound  tone  is,  for 

*  [A  very  convenient  form  of  this  experi-  The  tone  is  also  brighter  and  unaccompanied 

nient,  useful  even  for  lecture  purposes,  is  to  by  any  windrush.     By  pressing  the  handle  of 

employ  two  tuning-forks,  tuned  as  an  Octave,  the  deeper  fork  on  the  table,  we  can  excite  its 

say  (■'  and  c",  and  held  over  separate  resonance  other  upper  partials,  and  thus  produce  a  third 

jars.    By  removing  first  one  and  then  the  other,  quality  of  tone,  which  can  be  readily  apprc- 

or  letting  both  sound  together,  the  above  effects  elated;  thus,  simple  c',  simple  c'  -t-  simple  c", 

can  be  made  evident,  and  they  even  remain  compound  c'. — Translator.'] 
when  the  Octave  is  not  tuned  perfectly  true. 


62  ANALYSIS  OF  COMPOUND  SENSATIONS.  part  i. 

musical  purposes,  determined  by  that  of  its  prime,  the  influence  of  the  upper 
partial  tones  is  by  no  means  unfelt.  They  give  the  compound  tone  a  brighter  an  d 
higher  effect.  Simple  tones  are  dull.  When  they  are  compared  with  compound 
tones  of  the  same  pitch,  we  are  inclined  to  estimate  the  compound  as  belonging  to 
a  higher  Octave  than  the  simple  tones.  The  difference  is  of  the  same  kind  as  that 
heard  when  first  the  vowel  oo  in  too  and  then  a  in  tar  are  simg  to  the  same  note. 
It  is  often  extremely  difficult  to  compare  the  pitches  of  compound  tones  of  different 
qualities.  It  is  very  easy  to  make  a  mistake  of  an  Octave.  This  has  happened 
to  the  most  celebrated  musicians  and  acousticians.  Thus  it  is  well  known  that 
Tartini,  who  was  celebrated  as  a  violinist  and  theoretical  musician,  estimated  all 
combinational  tones  (Chap.  XI.)  an  Octave  too  high,  and,  on  the  other  hand, 
Henrici  *  assigns  a  pitch  too  low  by  an  Octave  to  the  upper  partial  tones  of 
tuning-forks.f 

H  The  pi'oblem  to  be  solved,  then,  in  distinguishing  the  partials  of  a  compound 
tone  is  that  of  analysing  a  given  aggregate  of  sensations  into  elements  which  no 
longer  admit  of  analysis.  We  are  accustomed  in  a  large  number  of  cases  whei*e 
sensations  of  different  kinds  or  in  different  parts  of  the  body,  exist  simultaneously, 
to  recognise  that  they  are  distinct  as  soon  as  they  are  perceived,  and  to  direct  our 
attention  at  will  to  any  one  of  them  separately.  Thus  at  any  moment  we  can  be 
separately  conscious  of  what  wc  see,  of  what  we  hear,  of  what  we  feel,  and  dis- 
tinguish what  we  feel  in  a  finger  or  in  the  great  toe,  whether  pressure  or  a  gentle 
touch,  or  warmth.  So  also  in  the  field  of  vision.  Indeed,  as  I  shall  endeavour  to  show 
in  what  follows,  we  readily  distinguish  our  sensations  from  one  another  when  we 
have  a  precise  knowledge  that  they  are  composite,  as,  for  example,  when  we  have 
become  certain,  by  frequently  repeated  and  invariable  experience,  that  our  present 
sensation  arises  from  the  simultaneous  action  of  many  independent  stimuli,  each 
of  which  usually  excites  an  equally  well-known  individual  sensation.     This  induces 

H  us  to  think  that  nothing  can  be  easier,  when  a  number  of  different  sensations  are 
simultaneously  excited,  than  to  distinguish  them  individually  from  each  other,  and 
that  this  is  an  innate  facility  of  our  minds. 

Thus  we  find,  among  others,  that  it  is  quite  a  matter  of  course  to  hear  sepa- 
rately the  different  musical  tones  which  come  to  our  senses  collectively,  and  expect 
that  in  every  case  when  two  of  them  occur  together,  we  shall  be  able  to  do  the 
like. 

The  matter  is  very  different  when  we  set  to  work  at  investigating  the  more  un- 
usual cases  of  perception,  and  at  more  completely  understanding  the  conditions  under 
which  the  above-mentioned  distinction  can  or  cannot  be  made,  as  is  the  case  in  the 
physiology  of  the  senses.  We  then  become  aware  that  two  different  kinds  or  grades 
must  be  distinguished  in  our  becoming  conscious  of  a  sensation.  The  lower  grade  of 
this  consciousness,  is  that  where  the  influence  of  the  sensation  in  question  makes 
itself  felt  only  in  the  conceptions  we  form  of  external  things  and  processes,  and  assists 

H  in  determining  them.  This  can  take  place  without  our  needing  or  indeed  being  able 
to  ascertain  to  what  particular  part  of  our  sensations  we  owe  this  or  that  relation 
of  our  perceptions.  In  this  case  we  will  say  that  the  impression  of  the  sensation  in 
question  is  p^}xeived  synthetically.  The  second  and  higher  grade  is  when  we 
immediately  distinguish  the  sensation  in  question  as  an  existing  part  of  the  sum 
of  the  sensations  excited  in  us.  We  will  say  then  that  the  sensation  is  perceived 
analytically. X     The  two  cases  must  be  carefully  distinguished  from  each  other. 

*  Poggd.    Aim.,   vol.    xcix.    p.    506.      The  with  ivahrgcnommen,  and  then  restricting  the 

same  difficulty   is   mentioned   by  Zamminer  meaning  of  this  very  common  German  word. 

{Die    Mtisik    und    die    niusikalischen    I/istru-  It  appeared  to  me  that  it  would  be  clearer  to 

m«nie,  1855,  p.  Ill)  as  well  known  to  musicians.  an  English  reader  not  to  invent  new  words 

+  [Here   the   passage   from  '  The  problem  or   restrict   the   sense   of   old   words,   but    to 

to   be   solved,'   p.    62h,    to    '  from   its    simple  use  perceived  in  both  cases,  and  distinguish 

tones,'  p.  65b,  is  inserted  in  this  edition  from  the  them  (for  percipirt  and  appercipirt  respectively) 

4th  German  edition. —  'Translator.']  by  the  adjuncts  syntlietically  and  analytically, 

X  [Prof.  Helmholtz  uses  Leibnitz's  terms  the  use  of  which  is  clear  from  the  explanations 

percipirt  and  appercipirt,  alternating  the  latter  given  in  the  text.- — Traiislator.] 


CHAP.  IV.  ANALYSIS  OF  COMPOUND  SENSATIONS.  Cy^^ 

Seebeck  and  Ohm  are  agreed  that  the  upper  partials  of  a  musical  tone  are 
perceived  synthetically.  This  is  acknowledged  by  Seebeck  when  he  admits  that 
their  action  on  the  ear  changes  the  force  or  quality  of  the  sound  examined.  The 
dispute  turns  upon  whether  in  all  cases  they  can  be  perceived  analytically  in  their 
individual  existence  ;  that  is,  whether  the  ear  when  unaided  by  resonators  or  other 
physical  auxiliaries,  which  themselves  alter  the  mass  of  musical  somid  heard  by  the 
observer,  can  by  mere  direction  and  intensity  of  attention  distinguish  wlicther,  and 
if  so  in  what  force,  the  Octave,  the  Twelfth,  etc.,  of  the  prime  exists  in  the  given 
musical  sound. 

In  the  first  place  I  will  adduce  a  series  of  examples  which  show  that  the 
difficulty  felt  iu  analysing  musical  tones  exists  also  for  other  senses.  Let  us 
begin  with  the  comparatively  simple  perceptions  of  the  sense  of  taste.  The 
ingredients  of  our  dishes  and  the  spices  with  which  we  flavour  them,  are  not  so 
complicated  that  they  could  not  be  readily  learned  by  any  one.  And  yet  there  are  ^ 
very  few  people  who  have  not  themselves  practically  studied  cookery,  that  are  able 
readily  and  correctly  to  discover,  by  the  taste  alone,  the  ingredients  of  the  dishes 
placed  before  them.  How  much  practice,  and  perhaps  also  peculiar  talent,  belongs 
to  wine  tasting  for  the  piu-pose  of  discovering  adulterations  is  known  in  all  wine- 
growing countries.  Similarly  for  smell ;  indeed  the  sensations  of  taste  and  smell 
may  unite  to  form  a  single  whole.  Using  our  tongues  constantly,  we  are  scarcely 
aware  that  the  peculiar  character  of  many  articles  of  food  and  drink,  as  vinegar  or 
wine,  depends  also  upon  the  sensation  of  smell,  their  vapours  entering  the  back 
part  of  the  nose  through  the  gullet.  It  is  not  till  we  meet  with  persons  in  whom 
the  sense  of  smell  is  deficient  that  we  learn  how  essential  a  part  it  plays  iu 
tasting.  Such  persons  are  constantly  in  fault  when  judging  of  food,  as  indeed  any 
one  can  learn  from  his  own  experience,  when  he  suffers  from  a  heavy  cold  in  the 
head  without  having  a  loaded  tongue. 

When  our  hand  glides  unawares  along  a  cold  and  smooth  piece  of  metal  we^ 
are  apt  to  imagine  that  we  have  wetted  our  hand.  This  shows  that  the  sensation 
of  wetness  to  the  touch  is  compounded  out  of  that  of  unresisting  gliding  and  cold, 
which  in  one  case  results  from  the  good  heat-conducting  properties  of  metal,  and 
in  the  other  from  the  cold  of  evaporation  and  the  great  specific  heat  of  water. 
We  can  easily  recognise  both  sensations  in  wetness,  when  we  think  over  the 
matter,  but  it  is  the  above-mentioned  illusion  which  teaches  us  that  the  peculiar 
feeling  of  wetness  is  entirely  resolvable  into  these  two  sensations. 

The  discovery  of  the  stereoscope  has  taught  us  that  the  power  of  seeing  the 
depths  of  a  field  of  view,  that  is,  the  different  distances  at  which  objects  and 
their  parts  lie  from  the  eye  of  the  spectator,  essentially  depends  on  the  simul- 
taneous synthetical  perceptions  of  two  somewhat  different  perspective  images  of 
the  same  objects  by  the  two  eyes  of  the  observer.  If  the  difference  of  the  tAvo 
images  is  sufficiently  great  it  is  not  difficult  to  perceive  them  analytically  as 
separate.  For  example,  if  we  look  intently  at  a  distant  object  and  hold  one  of  ^ 
our  fingers  slightly  in  front  of  our  nose  we  see  two  images  of  our  finger  against 
the  background,  one  of  which  vanishes  when  we  close  the  right  eye,  the  other 
belonging  to  the  left.  But  when  the  differences  of  distance  are  relatively  small, 
and  hence  the  differences  of  the  two  perspective  images  on  the  retina  are  so  also, 
great  practice  and  certainty  in  the  observation  of  double  images  is  necessary  to 
keep  them  asiuider,  yet  the  synthetical  perception  of  their  differences  still  exists, 
and  makes  itself  felt  in  the  apparent  relief  of  the  surface  viewed.  In  this  case 
also,  as  well  as  for  upper  partial  tones,  the  ease  and  exactness  of  the  analytical 
perception  is  far  behind  that  of  the  synthetical  perception. 

In  the  conception  which  we  form  of  the  direction  in  which  the  objects  viewed 
seem  to  lie,  a  considerable  part  must  be  played  by  those  sensations,  mainly  muscular, 
which  enable  us  to  recognise  the  position  of  our  body,  of  the  head  with  regard  to 
the  body,  and  of  the  eye  with  regard  to  the  head.  If  one  of  these  is  altered,  for 
example,  if  the  sensation  of  the  proper  position  of  the  eye  is  changed  by  pressing 


64  ANALYSIS  OF  COMPOUND  SENSATIONS.  part  i. 

a  finger  against  the  eyeball  or  by  injury  to  one  of  the  nuiscles  of  the  eye,  our  per- 
ception of  the  position  of  visible  objects  is  also  changed.  But  it  is  only  by  such 
occasional  illusions  that  we  become  aware  of  the  fact  that  muscular  sensations  form 
part  of  the  aggregate  of  sensations  by  which  our  conception  of  the  position  of  a 
visible  object  is  determined. 

The  phenomena  of  mixed  colours  preseiit  considerable  analogy  to  those  of  com- 
pound musical  tones,  only  in  the  case  of  colour  the  number  of  sensations  reduces  to 
three,  and  the  analysis  of  the  composite  sensations  into  their  simple  elements  is  still 
more  difficult  and  imperfect  than  for  musical  tones.  As  early  as  1686  R.  Waller 
mentions  in  the  Philosophical  Transactimis  the  reduction  of  all  colours  to  the 
mixture  of  three  fundamental  colours,  as  something  already  well  known.  This 
view  could  in  earlier  times  only  be  founded  on  sensations  and  experiments  arising 
from  the  mixture  of  pigments.     In  recent  times  we  have  discovered  better  methods, 

H  by  mixing  light  of  different  colours,  and  hence  have  confirmed  the  correctness  of 
that  hypothesis  by  exact  measurements,  but  at  the  same  time  we  have  learned  that 
this  confirmation  only  succeeds  within  a  certain  limit,  conditioned  by  the  fact  that  no 
kind  of  coloured  light  exists  which  can  give  us  the  sensation  of  a  single  one  of  the 
fundamental  colours  with  exclusive  purity.  Even  the  most  saturated  and  purest 
colours  that  the  external  world  presents  to  us  in  the  prismatic  spectrum,  may  by 
the  development  of  secondary  images  of  the  complementary  colours  in  the  eye 
be  still  freed  as  it  were  from  a  white  veil,  and  hence  cannot  be  considered  as  abso- 
lutel}^  pure.  For  this  reason  we  are  unable  to  show  objectively  the  absolutely  pure 
fundamental  colours  from  a  mixture  of  which  all  other  colours  without  exception 
can  be  formed.  We  only  know  that  among  the  colours  of  the  spectrum  scarlet-i^ed, 
yellow-green,  and  blue- violet  approach  to  them  nearer  than  any  other  oljjective 
colours.*  Hence  we  are  able  to  compound  out  of  these  three  colours  almost  all  the 
colours  that  usually  occur  in  different  natural  bodies,  but  we  cannot  produce  the 

H  yellow  and  blue  of  the  spectrum  in  that  complete  degree  of  saturation  which  they 
reach  when  purest  within  the  spectrum  itself.  Our  mixtures  are  always  a  little 
whiter  than  the  corresponding  simple  colours  of  the  spectrum.  Hence  it  follows 
that  we  never  see  the  simple  elements  of  our  sensations  of  colour,  or  at  least  see 
them  only  for  a  very  short  time  in  particular  experiments  directed  to  this  end,  and 
consequently  cannot  have  any  such  exact  or  certain  image  in  our  recollection,  as 
would  indisputably  be  necessary  for  accurately  analysing  every  sensation  of  colour 
into  its  elementary  sensations  by  inspection.  Moreover  we  have  relatively  rare 
opportunities  of  observing  the  process  of  the  composition  of  colours,  and  hence  of 
recognising  the  constituents  in  the  compound.  It  certainly  appears  to  me  very 
characteristic  of  this  process,  that  for  a  century  and  a  half,  from  Waller  to  Goethe, 
every  one  relied  on  the  mixtures  of  pigments,  and  hence  believed  green  to  be  a 
mixture  of  blue  and  yellow,  whereas  when  sky-blue  and  sulphur-yellow  beams  of 
light,    not  pigments,  are  mixed  together,    the  result  is  white.      To  this  very  cir- 

Hcumstance  is  due  the  violent  opposition  of  Goethe,  who  was  only  acquainted  with 
the  colours  of  pigments,  to  the  assertion  that  white  w^as  a  mixture  of  variously 
coloured  beams  of  light.  Hence  we  can  have  little  doubt  that  the  power  of  dis- 
tinguishing the  different  elementary  constituents  of  the  sensation  is  originally 
absent  in  the  sense  of  sight,  and  that  the  little  which  exists  in  highly  educated 
observers,  has  been  attained  by  specially  conducted  experiments,  through  which  of 
course,  when  wrongly  planned,  error  may  have  ensued. 

On  the  other  hand  every  individual  has  an  opportunity  of  experimenting  on  the 

*  [In  Ms  Physiological  Optics,  p.  227,  E,'  hence  I  translate  span-griin  by  'yellow- 
Prof.  Helmholtz  calls  scarlet-red  or  vermilion  green.'  Maxwell's  blue  or  third  colour  was 
the  part  of  the  spectrum  before  reaching  between  the  lines  F  and  G,  but  twice  as  far 
Fraunhofer's  line  C.  He  does  not  use  span-  from  the  latter  as  the  former,  This  gives  the 
g7'Un  {  =  Griui-span  or  verdigris,  literally  colour  which  Prof.  H.  in  his  0;y<a-s  calls  '  cya- 
'  Spanish-green ')  in  his  Optics,  but  talks  of  nogen  blue,'  or  Prussian  blue.  The  violet 
green-yellow  between  the  lines  E  and  b,  and  proper  does  not  begin  till  after  the  line  G.  It 
he  says,  on  p.  844,  that  Maxwell  took  as  one  of  is  usual  to  speak  of  these  throe  colours,  vaguely, 
the  fundamental  colours  '  a  green  near  the  line  as  Red,  Green,  and  Blue.  — Translator.] 


CHAPS.  IV.  V.  ANALYSIS  OF  COMPOUND  SENSATIONS.  65 

composition  of  two  or  more  musical  sounds  or  noises  on  the  most  extended  scale 
and  the  power  of  analysing  even  extremely  involved  compounds  of  musical  tones, 
into  the  separate  parts  produced  hy  individual  instruments,  can  readily  be  acquired 
by  any  one  who  directs  his  attention  to  the  subject.  But  the  ultimate  simple 
elements  of  the  sensation  of  tone,  simple  tones  themselves,  are  rarely  heard  alone. 
Even  those  instruments  by  which  they  can  be  prodiiced,  as  tuning-forks  before 
resonance  chambers,  when  strongly  excited,  give  rise  to  weak  harmonic  upper 
partials,  jjartly  within  and  partly  without  the  ear,  as  Ave  shall  see  in  Chapters  V. 
and  VII.  Hence  in  this  case  also,  the  opportunities  are  very  scanty  for  impress- 
ing on  our  memory  an  exact  and  sure  image  of  these  simple  elementary  tones. 
But  if  the  constituents  to  be  added  are  only  indefinitely  and  vaguely  known,  the 
analysis  of  the  sum  into  those  parts  must  be  correspondingly^  uncertain.  If  we  do 
not  know  with  certainty  how  much  of  the  musical  tone  under  consideration  is  to 
be  attributed  to  its  prime,  we  cannot  but  be  luicertain  as  to  what  belongs  to  the  H 
partials.  Consequently  we  must  begin  by  making  the  individual  elements  which 
have  to  be  distinguished,  individually  audible,  so  as  to  obtain  an  entirely  fresh 
recollection  of  the  corresponding  sensation,  and  the  whole  business  requires  un- 
disturbed and  concentrated  attention.  We  are  even  without  the  ease  that  can  be 
obtained  by  frequent  repetitions  of  the  experiment,  such  as  we  possess  in  the 
analysis  of  musical  chords  into  their  individual  tones.  In  that  case  we  hear  the 
individual  tones  sufficiently  often  by  tliemselves,  whereas  we  rarely  hear  simple 
tones  and  may  almost  be  said  never  to  hear  the  building  up  of  a  compound  from  its 
simple  tones. 

The  results  of  the  preceding  discussion  may  be  summed  up  as  follows  : — ■ 

(1)  The  upper  partial  tones  corresponding  to  the  simple  vibrations  of  a  com- 
pound motion  of  the  air,  are  perceived  synthetically,  even  when  they  are  not  always 
perceived  analytically. 

(2)  But  they  can  be  made  objects  of  analytical  perception  withoxit  any  other  U 
help  than  a  proper  direction  of  attention. 

(3)  Even  in  the  case  of  their  not  being  separately  perceived,  because  they  fuse 
into  the  whole  mass  of  musical  sound,  their  existence  in  our  sensation  is  established 
by  an  alteration  in  the  quality  of  tone,  the  impression  of  their  higher  pitch  being 
characteristically  marked  by  increased  brightness  and  acuteness  of  quality. 

In  the  next  chapter  we  shall  give  details  of  the  relations  of  the  upper  partials 
to  the  quality  of  compound  tones. 


CHAPTER  V. 

ON    THE    DIFFERENCES    IN    THE    QUALITY    OF    MUSICAL    TONES. 

Towards  the  close  of  Chapter  I  (p.  21d),  we  found  that  differences  in  the  quality 
of  musical  tones  must  depend  on  the  form  of  the  vibration  of  the  air.  The  H 
reasons  for  this  assertion  were  only  negative.  We  have  seen  that  force  depended 
on  amplitude,  and  pitch  on  rapidity  of  vibration :  nothing  else  was  left  to  distin- 
guish quality  but  vibrational  form.  We  then  proceeded  to  show  that  the  existence 
and  force  of  the  upper  partial  tones  which  accompanied  the  prime  depend  also  on 
the  vibrational  form,  and  hence  we  could  not  but  conclude  that  musical  tones  of 
the  same  quality  would  always  exhibit  the  same  combination  of  partials,  seeing 
that  the  peciiliar  vibrational  form  which  excites  in  the  ear  the  sensation  of  a  certain 
quality  of  tone,  must  always  evoke  the  sensation  of  its  corresponding  upper  partials. 
The  question  then  arises,  can,  and  if  so,  to  what  extent  can  the  difterences  of 
musical  quality  be  reduced  to  the  combination  of  difterent  partial  tones  with  dif- 
ferent intensities  in  different  musical  tones?  At  the  conclusion  of  last  chapter 
(p.  QOd),  we  saw  that  even  artificially  combined  simple  tones  were  capable  of  fusing 
into  a  musical  tone  of  a  quality  distinctly  different  from  that  of  either  of  its  con- 
stituents, and  that  consequently  the  existence  of  a  new  upper  partial  really  altered 

F 


66  CONCEPTION  OF  MUSICAL  QUALITY  AND  TONE.  part  i. 

the  quality  of  a  tone.     By  this  means  we  gained  a  clue  to  the  hitherto  enigmatical 
nature  of  quality  of  tone,  and  to  the  cause  of  its  varieties. 

There  has  been  a  general  inclination  to  credit  quality  with  all  possible  pecu- 
liarities of  musical  tones  that  were  not  evidently  due  to  force  and  pitch.  This  was 
correct  to  the  extent  that  quality  of  tone  was  merely  a  negative  conception.  But 
very  slight  consideration  will  suffice  to  show  that  many  of  these  peculiarities  of 
musical  tones  depend  upon  the  way  in  which  they  begin  and  end.  The  methods  of 
attacking  and  releasing  tones  are  sometimes  so  characteristic  that  for  the  human 
voice  they  have  been  noted  by  a  series  of  different  letters.  To  these  belong  the  ex- 
plosive consonants  B,  D,  G,  and  P,  T,  K.  The  effects  of  these  letters  are  produced 
by  opening  the  closed,  or  closing  the  open  passage  through  the  mouth.  For  B 
and  P  the  closure  is  made  by  the  lips,  for  D  and  T  by  the  tongue  and  upper  teeth,* 
for  G  and  K  by  the  back  of  the  tongue  and  soft  palate.     The  series  of  the  mediae 

U  B,  D,  G  is  distinguished  from  that  of  the  tenues  P,  T,  K,  by  the  glottis  being  suffi- 
ciently narrowed,  when  the  closure  of  the  former  is  released,  to  produce  voice,  or  at 
least  the  rustle  of  whisper,  whereas  for  the  latter  or  tenues  the  glottis  is  wide  open,t 
and  cannot  sound.  The  mediae  are  therefore  accompanied  by  voice,  which  is 
capable  of  counnencing  at  the  beginning  of  a  syllable  an  instant  before  the  open- 
ing of  the  mouth,  and  of  lasting  at  the  end  of  a  syllable  a  moment  after  the  closure 
of  the  mouth,  because  some  air  can  be  still  driven  into  the  closed  cavity  of  the 
mouth  and  the  vibration  of  the  vocal  chords  in  the  larynx  can  be  still  maintained. 
On  account  of  the  narrowing  of  the  glottis  the  influx  of  air  is  more  moderate,  and 
the  noise  of  the  wind  less  sharp  for  the  mediae  than  the  tenues,  which,  being  spoken 
with  open  glottis,  allow  of  a  great  deal  of  wind  being  forced  at  once  from  the  chest.;]: 
At  the  same  time  the  resonance  of  the  cavity  of  the  mouth,  which,  as  we  shall 
more  clearly  understand  further  on,  exercises  a  great  influence  on  the  vowels, 
varies  its  pitch,  corresponding  to  the  rapid  alterations  in  the  magnitude  of  its  volume 

H  and  orifice,  and  this  brings  about  a  corresponding  rapid  variation  in  the  quality  of 
the  speech  sound. 

As  with  consonants,  the  difterences  in  the  quality  of  tone  of  struck  strings, 
also  partly  depends  on  the  rapidity  with  which  the  tone  dies  away.  When  the 
strings  have  little  mass  (such  as  those  of  gut),  and  are  fastened  to  a  very  mobile 
sounding  board  (as  for  a  violin,  guitar,  or  zither),  or  when  the  parts  on  which  they 
rest  or  which  they  touch  are  but  slightly  elastic  (as  when  the  violin  strings,  for 
example,  are  pressed  on  the  finger  board  by  the  soft  point  of  the  finger),  their 
vibrations  rapidly  disappear  after  striking,  and  the  tone  is  dry,  short,  and  without 
ring,  as  in  the  pizzicato  of  a  violin.  But  if  the  strings  are  of  metal  wire,  and 
hence  of  greater  weight  and  tension,  and  if  they  are  attached  to  strong  heavy 
bridges  w^hich  cannot  be  much  shaken,  they  give  out  their  vibrations  slowly  to  the 

*  [This  is  true  for  German,  and  most  Con-  examples,  it  seemed  better  in  the  present  case, 

tinental   languages,   and   for    some    dialectal  where  the  author  was  speaking  especially  of 

U  English,  especially  in  Cumberland,  Westmore-  the  phenomena  of  speech  to  which  he  was 

land,  Yorkshire,  Lancashire,  the  Peak  of  Derby-  personally  accustomed,  to  leave  the  text  un- 

shire,  and  Ireland,  but  even  then  only  in  con-  altered  and  draw  attention  to  English  pecu- 

nection  with  the  trilled  R.     Throughout  Eng-  liarities  in  footnotes. — Translator.'] 

land  generally,  the  tip  of  the  tongue  is  quite  \  [Observe  again  that  this  description  of 

free  from  the  teeth,  except  for  TH  in  thin  and  the   rush   of   wind    accompanying   P,    T,    K, 

then,  and  for  T  and  D  it  only  touches  the  hard  although  true  for  German  habits  of  speech,  is 

palate,  seldom  advancing  so  far  as  the  root  of  not  true  for  the  usual  English  habits,  which 

the  gums. —  Translator.']  require  the  windrush  between  the  opening  of 

t  [This  again  is  true  for  German,  but  not  the  mouth  and  sounding  of  the  vowel  to  be 

for  English,  French,  or  Italian,  and  not  even  entirely  suppressed.     The  English  result  is  a 

for  the  adjacent  Slavonic  languages.     In  these  gliding  vowel  sound  preceding  the  true  vowel  on 

languages  the  glottis  is  quite  closed  for  both  commencing  a  syllable,  and  following  the  vowel 

the  mediae  and  the  tenues  in  ordinary  speech,  on  ending  one.  The  difference  between  English 

but   the   voice   begins   for   the   mediae  before  P  and  German  Pis  precisely  the  same  (as  I  have 

releasing  the  closure  of  the  lips  or  tongue  and  verified  by  actual  observation  i  a.^  ihat  between 

palate,  and  for  the  tenues  at  the  moncnt  of  the  simple  Sanscrit  tenuis  P,  and  the  postaspi- 

releasc.     Although  in  giving  vowel  sounds,  &c.,  rated  Sanscrit  Ph,  as  now  actually  pronounced 

I  have  generally  contented  myself  with  trans-  by  cultivated  Bengalese.    See  raj  Early  English 

lating   the   same   into   English   symbols  and  Pronunciation,  p.  1136,  col.  1. — TroMslaior.'] 


CHAP.  V.  CONCEPTION  OF  MUSICAL  QUALITY  AND  TONE.  67 

air  and  the  sounding  board ;  their  vibrations  continue  longer,  their  tone  is  more 
durable  and  fuller,  as  in  the  pianoforte,  but  is  comparatively  less  powerful  and 
penetrating  than  that  of  gut  strings,  which  give  up  their  tone  more  readily  when 
struck  with  the  same  force.  Hence  the  pizzicato  of  bowed  instruments  when  well 
executed  is  much  more  piercing  than  the  tone  of  a  pianoforte.  Pianofortes  with 
their  strong  and  heavy  supports  for  the  strings  have,  consequently,  for  the  same 
thickness  of  string,  a  less  penetrating  but  a  much  more  lasting  tone  than  those 
instruments  of  which  the  suppoi'ts  for  the  strings  are  lighter. 

It  is  very  characteristic  of  brass  instruments,  as  trumpets  and  trombones, 
that  their  tones  commence  abruptly  and  sluggishly.  The  various  tones  in  these 
instruments  are  produced  by  exciting  different  upper  partials  through  different 
styles  of  blowing,  which  serve  to  throw  the  column  of  air  into  vibrating  portions 
of  different  numbers  and  lengths  similar  to  those  on  a  string.  It  always  requires 
a  certain  amount  of  effort  to  excite  the  new  condition  of  vibration  in  place  of  the  H 
old,  but  when  once  established  it  is  maintained  with  less  exertion.  On  the  other 
hand,  the  transition  from  one  tone  to  another  is  easy  for  wooden  wind  instruments, 
as  the  flute,  oboe,  and  clarinet,  where  the  length  of  the  column  of  air  is  readily 
changed  by  application  of  the  fingers  to  the  side  holes  and  keys,  and  where  the 
style  of  blowing  has  not  to  be  materially  altered. 

These  examples  will  suffice  to  show  how  certain  characteristic  peculiarities  in 
the  tones  of  several  instruments  depend  on  the  mode  in  which  they  begin  and  end. 
When  we  speak  in  what  follows  of  musical  quality  of  tone,  we  shall  disregard 
these  peculiarities  of  beginning  and  ending,  and  confine  our  attention  to  the 
peculiarities  of  the  musical  tone  which  continues  uniformly. 

But  even  when  a  musical  tone  continues  with  uniform  or  variable  intensity, 
it  is  mixed  up,  in  the  general  methods  of  excitement,  with  certain  noises,  which 
express  greater  or  less  irregularities  in  the  motion  of  the  air.  In  wind  instruments 
where  the  tones  are  maintained  by  a  stream  of  air,  we  generally  hear  more  or  less  H 
whizzing  and  hissing  of  the  air  which  breaks  against  the  sharp  edges  of  the 
mouthpiece.  In  strings,  rods,  or  plates  excited  by  a  violin  bow,  we  usually  hear 
a  good  deal  of  noise  from  the  rubbing.  The  hairs  of  the  bow  are  naturally  full  of 
many  minute  irregularities,  the  resinous  coating  is  not  spread  over  it  with  absolute 
evenness,  and  there  are  also  little  inequalities  in  the  motion  of  the  arm  which 
holds  the  bow  and  in  the  amount  of  pressure,  all  of  which  influence  the  motion 
of  the  string,  and  make  the  tone  of  a  bad  instrument  or  an  unskilful  performer 
rough,  scraping,  and  variable.  We  shall  not  be  able  to  explain  the  nature  of  the 
motions  of  the  air  and  sensations  of  the  ear  which  correspond  to  these  noises  till 
we  have  investigated  the  conception  of  heats.  Those  who  listen  to  music  make 
themselves  deaf  to  these  noises  by  purposely  withdrawing  attention  from  them,  but 
a  slight  amount  of  attention  generally  makes  them  very  evident  for  all  tones  pro- 
duced by  blowing  or  rubbing.  It  is  well  known  that  most  consonants  in  human 
speech  are  characterised  by  the  maintenance  of  similar  noises,  as  F,  V ;  S,  Z ;  TH  H 
in  thin  and  in  then ;  the  Scotch  and  German  guttural  CH,  and  Dutch  0.  For 
some  the  tone  is  made  still  more  irregular  by  trilling  parts  of  the  mouth,  as  for 
R  and  L.  In  the  case  of  R  the  stream  of  air  is  periodically  entirely  interrupted  by 
trilling  the  uvula*  or  the  tip  of  the  tongue ;  and  we  thus  obtain  an  intermitting 
sound  to  which  these  interruptions  give  a  ijeculiar  jarring  character.  In  the  case 
of  L  the  soft  side  edges  of  the  tongue  are  moved  by  the  stream  of  air,  and,  withoiit 
completely  interrupting  the  tone,  produce  inequalities  in  its  strength. 

Even  the  vowels  themselves  are  not  free  from  such  noises,  although  they  are 
kept  more  in  the  background  by  the  musical  character  of  the  tones  of  the  voice. 
Donders  first  drew  attention  to  these  noises,  which  are  partly  identical  with  those 
which  are  produced  when  the  corresponding  vowels  are  indicated  in  low  voiceless 

*  [In  the  northern  parts  of  Germany  and  of  There  are  also  many  other  trills,  into  which, 
France,  and  in  Northumberland,  but  not  other-  as  into  other  phonetic  details,  it  is  not  neces- 
wise  in  England,  except  as  an  organic  defect.       sary  to  enter. —  Translator.'] 

F    2 


68  CONCEPTION  OF  MUSICAL  QUALITY  AND  TONE.  part  r. 

speecli.  They  are  strongest  for  ee  in  see,  the  French  u  in  vu  (which  is  nearly  tlie 
same  as  the  Norfolk  and  Devon  oo  in  too),  and  for  oo  in  too.  For  these  vowels  they 
can  be  made  audible  even  when  speaking  aloud.*^  By  simply  increasing  their  force 
the  vowel  ee  in  see  becomes  the  consonant  y  in  yon,  and  the  vowel  oo  in  too  the 
consonant  w  in  wan.-\  For  a  in  art,  a  in  at,  e  in  met,  there,  and  o  in  more,  the 
noises  appear  to  me  to  be  produced  in  the  glottis  alone  when  speaking  gently,  and 
to  be  absorbed  into  the  voice  when  speaking  aloud.:}:  It  is  remarkable  that  in 
speaking,  the  vowels  a  in  art,  a  in  at,  and  e  in  met,  there,  are  produced  with  less 
musical  tone  than  in  singing.  It  seems  as  if  a  feeling  of  greater  compression  in 
the  larynx  caused  the  tuneful  tone  of  the  voice  to  give  way  to  one  of  a  more  jarring 
character  which  admits  of  more  evident  articulation.  The  greater  intensity  thus 
given  to  the  noises,  appeal's  in  this  case  to  facilitate  the  characterisation  of  the 
peculiar  vowel  quality.     In  singing,  on  the  contrary,  we  try  to  favour  the  musical 

U  part  of  its  quality  and  hence  often  render  the  articulation  somewhat  obscui-e.§ 

Such  accompanying  noises  and  little  inequalities  in  the  motion  of  the  air, 
furnish  much  that  is  characteristic  in  the  tones  of  musical  instruments,  and  in  the 
vocal  tones  of  speech  which  correspond  to  the  different  positions  of  the  mouth  ; 
but  besides  these  there  are  numerous  peculiarities  of  quality  belonging  to  the 
musical  tone  proper,  that  is,  to  the  perfectly  regular  portion  of  the  motion  of  the 
air.  The  importance  of  these  can  be  better  appreciated  by  listening  to  musical 
instruments  or  human  voices,  from  such  a  distance  that  the  comparatively  weaker 
noises  are  no  longer  audible.  Notwithstanding  the  absence  of  these  noises,  it  is 
generally  possible  to  discriminate  the  different  musical  instruments,  although  it 
must  be  acknowledged  that  under  such  circumstances  the  tone  of  a  French  horn 
may  be  occasionally  mistaken  for  that  of  the  singing  voice,  or  a  violoncello  may 
be  confused  with  an  harmonium.  For  the  human  voice,  consonants  first  disappear 
at  a  distance,  because  they  are  characterised  by  noises,  but  M,  N,  and  the  vowels 

f  can  be  distinguished  at  a  greater  distance.  The  formation  of  M  and  N  in  so  far 
resembles  that  of  vowels,  that  no  noise  of  wind  is  generated  in  any  part  of  the 
cavity  of  the  mouth,  which  is  perfectly  closed,  and  the  sound  of  the  voice  escapes 
through  the  nose.  The  mouth  merely  forms  a  resonance  chamber  which  alters  the 
quality  of  tone.  It  is  interesting  in  calm  weather  to  listen  to  the  voices  of  men 
who  are  descending  from  high  hills  to  the  plain.  Words  can  no  longer  be  recog- 
nised, or  at  most  only  such  as  are  composed  of  M,  N,  and  vowels,  as  Mamma,  Ko, 
Noon.  But  the  vowels  contained  in  the  spoken  words  are  easily  distinguished. 
Wanting  the  thread  which  connects  them  into  words  and  sentences,  they  form  a 
strange  series  of  alternations  of  quality  and  singular  inflections  of  tone. 

In  the  present  chapter  we  shall  at  first  disregard  all  irregular  portions  of  the 
motion  of  the  air,  and  the  mode  in  which  sounds  commence  or  terminate,  directing 
our  attention  solely  to  the  musical  part  of  the  tone,  properly  so  called,  which 
corresponds  to    a   uniformly   sustained    and  regularly  periodic   motion   of  the  air, 

^  and  we  shall  endeavour  to  discover  the  relations  between  the  quality  of  the  sound 

*  [At  the  Comedie  Fran(,'aise  I  have  heard  the  important   phonetic   observations  in  the 

M.   Got  pronounce   the  word  oui  and   Mme.  text. — Translator.] 

Provost-Ponsin  pronounce  the  last  syllable  of  §  [These   observations   must   not    be   cou- 

haehis  entirely  without  voice  tones,  and  yet  sidered  as  exhausting  the  subject  of  the  dif- 

make  them  audible  throughout  the  theatre. —  ference  between  the   singing  and  the   speak- 

Translator.]  ing   voice,    which   requires   a  peculiar   study 

t  [That  this  is  not  the  whole  of  the  pheno-  here  merely  indicated.     See  my  Pronunciatimi 

menon  is  shown  by  the  words  ye,  v^oo.     The  fur   Singers    (Curwen)    and     S-jxcch    in    Soncj 

whole  subject  is  discussed  at  length  in  my  (Novello).  The  difference  between  English  and 

Early  English    Prommciation,   pp.    1092-1094,  German  habits  of  speaking  and  singing  must 

and  1149-1151 — Translator.]  also   be   borne   in   mind,  and  allowed  for  by 

%  [By    '  speaking  gently '  [leise)   seems   to  the  reader.     The  English  vowels  given  in  the 

be  meant  either  speaking  absolutely  without  text  are  not  the  perfect  equivalents  of  Prof, 

voice,   that  is  with  an  open  glottis,  or  in  a  Helmholtz's    German    sounds.      The    noises 

whisper,  with  the  glottis  nearly  closed.     For  which  accompany  the  vowels  are  not  nearly 

voice  the  glottis  is  quite  closed,  and  this  is  so  marked  in  English  as  in  German,  but  they 

indicated   by   '  speaking   aloud '    {hcim   lantcn  differ  very  much  locally,  even  in  England. — 

Sprechen).     It  would  lead   too   far  to  discuss  Translator.] 


CHAP.  V.  1.  TONES  WITH  XO  UPPER  PARTIALS.  69 

and  its  composition  out  of  individual  simple  tones.     The  peculiarities  of  ([uality 
of  sound  belonging  to  this  division,  we  shall  briefly  call  its  musical  qua/it i/. 

The  object  of  the  present  chapter  is,  therefore,  to  describe  the  difl;creut  com- 
position of  musical  tones  as  produced  by  diflferent  instruments,  for  the  purpose  of 
showing  how  different  modes  of  combining  the  upper  partial  tones  correspond  to 
characteristic  varieties  of  musical  quality.  Certain  general  rules  will  result  for 
the  arrangement  of  the  upper  partials  which  answer  to  such  species  of  musical 
quality  as  are  called,  soft,  jnercinff,  fmii/mg,  hoUoic  or  poor,  full  or  rich,  dull, 
hright,  cfisp,  jmngent,  and  so  on.  Independently  of  our  immediate  object  (the 
determination  of  the  physiological  action  of  the  ear  in  the  discrimination  of 
musical  quality,  which  is  reserved  for  the  following  chapter),  the  results  of  this 
investigation  are  important  for  the  resolution  of  purely  musical  (piestions  in  later 
chapters,  because  they  show  us  how  rich  in  upper  partials,  good  musical  qualities 
of  tone  are  found  to  be,  and  also  point  out  the  peculiarities  of  musical  quality^ 
favoured  on  those  musical  instruments,  for  which  the  quality  of  tone  has  been  to 
some  extent  abandoned  to  the  caprice  of  the  maker. 

Since  physicists  have  worked  comparatively  little  at  this  subject  1  shall  be 
forced  to  enter  somewhat  more  minutely  into  the  mechanism  by  which  the  tones 
of  several  instruments  are  produced,  than  will  be,  perhaps,  agreeable  to  many  of 
my  readers.  For  such  the  principal  results  collected  at  the  end  of  this  chapter  will 
suffice.  On  the  other  hand,  I  must  ask  indulgence  for  leaving  many  large  gaps 
in  this  almost -unexplored  region,  and  for  confining  myself  principally  to  instru- 
ments sufficiently  well  known  for  us  to  obtain  a  tolerably  satisfactory  view  of  the 
source  of  their  tones.  In  this  inquiry  lie  rich  materials  -  for  interesting  acovistical 
work.  But  I  have  felt  bound  to  confine  myself  to  what  was  necessary  for  the 
continuation  of  the  present  investigation. 

1.  Musical  Tones  withoxit  Upper  Partials.  ^ 

We  begin  with  such  musical  tones  as  are  not  decomposable,  but  consist  of  a 
single  simple  tone.  These  are  most  readily  and  purely  produced  by  holding  a 
struck  tuning-fork  over  the  mouth  of  a  resonance  tube,  as  has  been  described  in 
the  last  chapter  (p.  54^/).*  These  tones  are  uncommonly  soft  and  free  from  all 
shrillness  and  roughness.  As  already  remarked,  they  appear  to  lie  comparatively 
deep,  so  that  such  as  correspond  to  the  deep  tones  of  a  bass  voice  produce  the 
impression  of  a  most  remarkable  and  unusual  depth.  The  musical  quality  of  such 
deep  simple  tones  is  also  rather  dull.  The  simple  tones  of  the  soprano  pitch 
sound  bright,  but  even  those  corresponding  to  the  highest  tones  of  a  soprano  voice 
are  very  soft,  without  a  trace  of  that  cutting,  rasping  shrillness  which  is  displayed 
by  most  instruments  at  such  pitches,  with  the  exception,  perhaps,  of  the  flute,  for 
which  the  tones  are  very  nearly  simple,  being  accompanied  with  very  few  and 
faint  upper  partials.  Among  vowels,  the  oo  in  too  comes  nearest  to  a  simple  tone, 
but  even  this  vowel  is  not  entirely  free  from  upper  partials.  On  comparing  the  H 
musical  quality  of  a  simple  tone  thus  produced  with  that  of  a  compound  tone  in 
which  the  first  harmonic  upper  partial  tones  are  developed,  the  latter  will  be  found 
to  be  more  tuneful,  metallic,  and  brilliant.  Even  the  vowel  oo  in  too,  although 
the  dullest  and  least  tuneful  of  all  vowels,  is  sensibly  more  brilliant  and  less  dull 
than  a  simple  tone  of  the  same  pitch.  The  series  of  the  first  six  partials  of  a 
compound  tone  may  be  regarded  musically  as  a  major  chord  with  a  very  predominant 
fundamental  tone,  and  in  fact  the  musical  quality  of  a  compound  tone  possessing 
these  partials,  as,  for  example,  a  fine  singing  voice,  when  heard  beside  a  simple  tone, 
very  distinctly  produces  the  agreeable  ettect  of  a  consonant  chord. 

Since  the  form  of  simple  waves  of  known  periodic  time  is  completely  given 
when  their  amplitude  is  given,   simple   tones  of   the   same  pitch   can   only  difter 
in  force  and  not  in  musical  quality.     In  fact,  the  difference  of  quality  remains 
*  On  possible  sources  of  disturbance,  see  Appendix  IV. 


70  TONES  WITH  INHARMONIC  UPPER  PARTIALS.  part  i. 

perfectly  indistinguishable,  whether  the  simple  tone  is  conducted  to  the  external 
air  in  the  preceding  methods  by  a  tuning-fork  and  a  resonance  tube  of  any  given 
material,  glass,  metal,  or  pasteboard,  or  by  a  string,  provided  only  that  we  guard 
against  any  chattering  in  the  apparatus. 

Simple  tones  accompanied  only  by  the  noise  of  rushing  wind  can  also  be  pro- 
duced, as  already  mentioned,  by  blowing  over  the  mouth  of  bottles  with  necks 
(p.  60c).  If  we  disregard  the  friction  of  the  air,  the  proper  musical  cpiality  of  such 
tones  is  really  the  same  as  that  produced  by  tuning-forks. 

2.   Musical  I'ones  with  Inharmonic  Up'per  Partlah. 

Nearest  to  musical  tones  without  any  upper  partials  are  those  with  secondary 
tones  which  are  inharmonic  to  the  prime,  and  such  tones,  therefore,  in  strictness, 
^should  not  be  reckoned  as  musical  tones  at  all.  They  are  exceptionally  used  in 
artistic  music,  but  only  when  it  is  contrived  that  the  prime  tone  should  be  so  much 
more  powerful  than  the  secondary  tones,  that  the  existence  of  the  latter  may  be 
ignored.  Hence  they  are  placed  here  next  to  the  simple  tones,  because  musically 
they  are  available  only  for  the  more  or  less  good  simple  tones  which  they  represent. 
The  first  of  these  are  tuning-forks  themselves,  when  they  are  struck  and  applied 
to  a  sounding  board,  or  brought  very  near  the  ear.  The  [inharmonic]  upper  partials 
of  tuning-forks  lie  very  high.  In  those  which  I  have  examined,  the  first  made 
from  5-8  to  6-6  as  many  vibrations  in  the  same  time  as  the  prime  tone,  and  hence 
lay  between  its  third  diminished  Fifth  and  major  Sixth.  The  pitch  numbers  of 
these  high  upper  partial  tones  were  to  one  another  as  the  squares  of  the  odd 
numbers.  In  the  time  that  the  first  upper  partial  would  execute  3x3  =  9  vibra- 
tions, the  next  would  execute  5x5  =  25,  and  the  next  7  x  7  =  49,  and  so  on.  Their 
pitch,   therefore,   increases  with   extraoi'dinary  rapidity,   and  they   are   usually   all 

^inharmonic  with  the  prime,  though  some  of  them  may  exceptionally  become 
harmonic.  If  we  call  the  prime  tone  of  the  fork  c,  the  next  succeeding  tones  are 
nearly  a"\),  cV,  c'it.*  These  high  secondary  tones  produce  a  bright  inharmonic 
clink,  which  is  easily  heard  at  a  considerable  distance  when  the  fork  is  first  struck, 
whereas  when  it  is  brought  close  to  the  ear,  the  prime  tone  alone  is  heard.  The 
ear  readily  separates  the  prime  from  the  upper  tones  and  has  no  inclination  to  fuse 
them.  The  high  simple  tones  usually  die  off  rapidly,  while  the  prime  tone  remains 
audible  for  a  long  time.  It  should  be  remarked,  however,  that  the  nuitual  relations 
of  the  proper  tones  of  tuning-forks  differ  somewhat  according  to  the  form  of  the 
fork,  and  hence  the  above  indications  must  be  looked  upon  as  merely  approximate. 
In  theoretical  determinations  of  the  upper  partial  tones,  each  prong  of  the  fork 
may  be  regarded  as  a  rod  fixed  at  one  end. 

The  same  relations  hold  for  straight  elastic  rods,  which,  as  already  mentioned, 
when  struck,  give  rather  high  inharmonic  upper  partial  tones.     When  such  a  rod 

f  is  firmly  supported  at  the  two  nodal  lines  of  its  prime  tone,  the  continuance  of 
that  tone  is  favo\ired  in  preference  to  the  other  higher  tones,  and  hence  the  latter 
disturb  the  effect  very  slightly,  more  especially  as  they  rapidly  die  away  after  the 
rod  has  been  struck.     Such  rods,  however,  are  not  suitable  for  real  artistic  music, 

*  [On  calculating  the  number  of  cents  (as  hence  it  is  called  d'^'  in  the  text.    The  interval 

in  App.  XX.  sect.  C),  we  find  tliat  the  first  to   the   next   tone   is   25  :  49  or  1165  cents. 

tone  mentioned,  which  vibrates  from  5-8  to  Adding  this  to  the  former  numbers  the  uiterval 

6-6  as  fast  as  tlie  prime,  makes  an  interval  with  the  prime  must  be  between  5977   and 

with  it  of  from  .3043  to  3267  ct.,  so  that  if  C201  cents,  or  between  ?/^'  +  77  and  (f '  -  3,  for 

the  prime  is  called  c,  the  note  lies  between  which  in  the  text  c"Jf  is  selected.     The  inde- 

g"\y  +  43,  and  a"  -  33,  where  <i"r,  and  a"  are  terminacy  arises  from  the  difftculty  of  finding 

the  third  diminished  Fifth  and  major  Sixth  of  the  pitch  of  the  first  inharmonic  upper  partial, 

the  prime  c  mentioned  in  the  text.     This  Prof.  The  intervals  between  that  and  the  next  upper 

Helmholtz  calls  a"%  or  3200  cents.    Then  the  partials  are  9  :  25  or  1769  ct.,  9  :  49  or  2934 

interval  between  this  partial  and  the  next  is  ct.,  9  :  81  or  3699  ct.,  and  so  on.     The  word 

9  :  25   or   1769   ct. ,   and   hence   the    interval  '  inharmonic  '  has  been  inserted  m  the  text, 

with    the   prime   is   between   4812   and   5036  as  tuning-forks  have  also  generally  harmonic 

cents,  or  lies  between  c"' +  12  and  d"'  +  36,  and  upper  partials.    See  p.  54(7,  noie.^Trunslatoi:'] 


CHAP.  V.  2.        TONES  WITH  INHARMONIC  UPPER  PARTIALS.  71 

although  tlicy  have  hitely  been  introduced  for  military  and  dance  music  on  accoimt 
of  their  penetrating  qualities  of  tone.  Glass  rods  or  plates,  and  wooden  rods,  were 
foi'inei'ly  used  in  this  way  for  the  <7/a.s.s  harnionicon  and  the  xti'dir-ficldle  or  icocxl- 
iMvmonicon.  Tlie  rods  were  inserted  between  two  pairs  of  intertwisted  strings^ 
which  grasped  them  at  their  two  nodal  lines.  The  wooden  rods  in  the  German 
strmr-jiddle  were  simply  laid  on  straw  cylinders.  'I'hey  were  struck  with  hanmiers 
of  wood  or  cork. 

The  only  eflect  of  the  material  of  the  rods  on  the  quality  of  tone  in  these 
cases,  consists  in  the  greater  or  less  length  of  time  that  it  allows  the  proper  tones, 
at  difterent  pitches  to  continue.  These  secondary  tones,  including  the  higher  ones, 
usually  continue  to  sound  longest  in  elastic  metal  of  fine  uniform  consistency, 
because  its  greater  mass  gives  it  a  greater  tendency  to  continue  in  any  state  of 
motion  which  it  has  once  assumed,  and  among  metals  the  most  perfect  elasticity 
is  found  in  steel,  and  the  better  alloys  of  copper  and  zinc,  or  copper  and  tin.  InH 
slightly  alloyed  precious  metals,  their  greater  specific  gravity  lengthens  the  dura- 
tion of  the  tone,  notwithstanding  their  inferior  elasticity.  Superior  elasticity 
a])pears  to  favoiu-  the  continuance  of  the  higher  proper  tones,  because  imperfect 
elasticity  and  friction  generally  seems  to  damp  rapid  more  (piickly  than  slow  vibra- 
tions. Hence  I  think  that  I  may  describe  the  general  characteristic  of  what  is. 
usually  called  a  nietallic  quality  of  tone,  as  the  comparatively  continuous  and 
uniform  maintenance  of  higher  upper  partial  tones.  The  quality  of  tone  for  glass 
is  similar  :  but  as  it  breaks  when  violently  agitated,  the  tone  is  always  weak  and 


soft,  and  it  is  also  comparatively  high,  and  dies  rapidly  away,  on  account  of  the 
smaller  mass  of  the  vibrating  body.  In  wood  the  mass  is  small,  the  internal 
structure  comparatively  rough,  being  full  of  countless  interstices,  and  the  elasticity 
also  comparatively  imperfect,  so  that  the  proper  tones,  especially  the  higher  ones, 
rapidly  die  away.  And  for  this  reason  the  straw-fiddle  or  wood  harmonicon  is  per- 
haps more  satisfactory  to  a  musical  ear,  than  harmonicons  formed  of  steel  or  glass 
rods  or  plates,  with  their  piercing  inharmonic  upper  partial  tones,— at  least  so  far 
as  simple  tones  are  suitable  for  music  at  all,  of  which  I  shall  have  to  speak  later  on. "■= 

For  all  of  these  instruments  which  have  to  be  struck,  the  hammers  are  made 
of  wood  or  cork,  and  covered  with  leather.  This  renders  the  highest  \ipper 
partials  much  weaker  than  if  only  hard  metal  hammers  were  employed.  Greater  H 
hardness  of  the  striking  mass  produces  greater  discontinuities  in  the  original 
motion  of  the  plate.  The  influence  exerted  by  the  manner  of  striking  will  be 
considered  more  in  detail,  in  reference  to  strings,  where  it  is  also  of  much  impor- 
taiice. 

Acconling  to  Ghladni's  discoveries,  elasitic  plates,  cut  in  circular,  oval,  scpiare, 
o])long,  triangular,  or  hexagonal  forms,  will  sound  in  a  great  numV)er  of  diff"erent 
vibrational  forms,  usually  producing  simple  tones  which  are  mutually  inharmonic. 
Fig.  21  gives  the  iriore  simple  vibrational  forms  of  a  circular  plate.  Much  more 
complicated  forms  occur  when  several  circles  or  additional  diameters  appear  as 
nodal  lines,  or  where  both  circles  and  diameters  occur.  Supposing  the  vibrational 
form  A  to  give  the  tone  c,  the  others  give  the  following  proper  tones  : — 

*  [In  Java  the  principal  music  is  produced  the  rods  are  laid  on  the  edges  of  boat-shaped 

by  harmonicons  of  metal  or  wooden  rods  and  vessels,  like  old  fashion  cheese-trays,  and  kept 

kettle-shaped  gongs.  The  wooden  harmonicons  in  position  by   nails  passing  loosely  through 

are  frequent  also  in  Asia  and  Africa.     In  Java  holes.     See  App.  XX.  sect.  K.— Translator.] 


72 


TONES  WITH  INHARMONIC  UPPER  PARTIALS. 


Number 
of  Nodal 
Circles 

Number  of  Diameters 

0 

1 

2                    3          j          4 

5 

0 

1 
2 

f'h 

^'b 

C          1         d' 

y"         1 

[ 

c"         j 

1 

'f-o"^ 

This  shoAvs  that  many  proper  tones  of  nearly  the  same  pitch  are  produced  by  a 
plate  of  this  kind.  When  a  plate  is  struck,  those  proper  tones  which  have  no 
node  at  the  point  struck,  will  all  sound  together.  To  obtain  a  particular  deter- 
minate tone  it  is  of  advantage  to  support  the  plate  in  points  which  lie  in  the  nodal 
lines  of  that  tone ;  because  those  proper  tones  which  have  no  node  in  those  points 
will   then  die  off  more   rapidly.     For  example,  if  a  circular  plate  is  supported  at 

H  3  points  in  the  nodal  circle  of  fig.  21,  C  (p.  71c),  and  is  struck  exactly  in  its  middle, 
the  simple  tone  called  (A  in  the  table,  which  belongs  to  that  form,  will  be  heard, 
and  all  those  other  proper  tones  which  have  diameters  as  some  of  their  nodal 
lines*  will  be  very  weak,  for  example,  c,  d',  c",  g",  b'\)  in  the  table.  In  the  same 
way  the  tone  g"^  with  two  nodal  circles,  dies  off  immediately,  because  the  points 
of  support  fall  on  one  of  its  ventral  segments,  and  the  first  proper  tone  which  can 
sound  loudly  at  the  same  time  is  that  corresponding  to  three  nodal  circles,  one  of 
its  nodal  lines  being  near  to  that  of  No.  2.  But  this  is  3  Octaves  and  more  than 
a  whole  Tone  higher  than  the  proper  tone  of  No.  2,  and  on  account  of  this  great 
interval  does  not  disturb  the  latter.  Hence  a  disc  thus  struck  gives  a  tolerably 
good  musical  tone,  whereas  plates  in  general  j)roduce  sounds  composed  of  many  in- 
harmonic proper  tones  of  nearly  the  same  pitch,  giving  an  empty  tin-kettle  sort  of 
quality,  which  cannot  be  used  in  music.  But  even  when  the  disc  is  ])roperly  sup- 
ported  the  tone   dies  away  rapidly,   at  least  in  the   case  of  glass  plates,  because 

^  contact  at  many  points,  even  when  nodal,  sensibly  impedes  the  freedom  of  vibra- 
tion. 

The  sound  of  hells  is  also  accompanied  by  inharmonic  secondary  tones,  which, 
however,  do  not  lie  so  close  to  one  another  as  those  of  flat  plates.  The  vibrations 
which  usually  arise  have  4,  6,  8,  10,  ifec,  nodal  lines  extending  from  the  vertex  of 
the  bell  to  its  margin,  at  equal  intervals  from  each  other.  The  corresponding 
proper  tones  for  glass  bells  which  have  approximativcly  the  same  thickness 
throughout,  are  nearly  as  the  squares  of  the  numbers  2,  3,  4,  5,  so  that  if  we  call 
the  lowest  tone  <\  wo  have  for  the 


Number  of  nodal  Hues  . 

.    1        4 

1 

G 

8 

10 

12 

Tones    

Cenfe 

•    1        ^' 
0 

d'  + 
1404 

c"      ' 
2400 

3173 

d"'  + 
3804 

The  tones,  however,  vary  with  the  greater  or  less  thickness  of  the  wail  of  the 
Hbell  towards  the  margin,  and  it  appears  to  be  an  essential  point  in  the  ait  of 
casting  bells,  to  make  the  deeper  proper  tones  mutually  harmonic  by  giving  the 
bell  a  certain  empirical  form.  According  to  the  observations  of  the  organist 
Gleitz.t  the  bell  cast  for  the  cathedral  at  Erfurt  in  1477  has  the  following  proper 
tones  :  E,  e,  g^,  h,  f-',  r/'jj,  //,  c"|.  The  [former]  bell  of  St.  Paul's,  London,  gave 
a  and  c'lf.  Hemony  of  Ziitphen,  a  master  in  the  seventeenth  century,  required  a 
good  bell  to  have  three  Octaves,  two  Fifths,  one  major  and  one  minor  Third.  The 
deepest  tone  is  not  the  strongest.  The  body  of  the  bell  when  struck  gives  a 
deeper  tone  than  the  'sound  bow,'  but  the  latter  gives  the  loudest  tone.  Probably 
other  vibrational  forms  of  bells  are  also  possible  in  which  nodal  circles  are  formed 


*  Provided  that  the  supported  points  do 
not  happen  to  belong  to  a  system  of  diameters 
making  equal  angles  with  each  other. 

t '  Historical  Notes  on  the  Great  Bell 
and   the   other   Bells    in    Erfurt   Cathedral' 


{Geschichtliche-s  iibcr  die  grosse  Glockc  und 
die  ilbrigen  Glocken  des  Domes  zu  Erfurt). 
Erfurt,  1867.— See  also  Schafh;iutl  in  the 
Kunst  und  Gewerbcblatt  fur  das  Konigreich 
Bay  cm,  1868,  liv.  325  to  350  ;  385  to  427. 


CHAP.  V.   Z. 


tonp:s  with  inharmonic  upper  partials. 


73 


parallel  to  the  margin.  But  these  seem  to  be  produced  with  ditliculty  autl  have 
not  yet  been  examined. 

If  a  bell  is  not  perfectly  symmetrical  in  respect  to  its  axis,  if,  for  example,  the 
wall  is  a  little  thicker  at  one  point  of  its  circumference  than  at  another,  it  will 
give,  on  being  struck,  two  different  tones  of  very  nearly  the  same  pitch,  which  will 
''beat'  together.  Four  points  on  the  margin  will  be  found,  separated  from  each 
other  by  quarter-circles,  in  which  only  one  of  these  tones  can  be  heard  without 
accompanying  beats,  and  four  others,  half-way  between  the  pairs  of  the  others, 
where  the  second  tone  only  sounds.  If  the  l)ell  is  struck  elsewhere  both  tones  are 
heard,  producing  beats,  and  such  beats  may  be  perceived  in  most  bells  as  their 
tone  dies  gradually  away. 

Stretclied  membranes  have  also  inharmonic  proper  tones  of  nearly  the  same 
pitch.  For  a  circular  membrane,  of  which  the  deepest  tone  is  c,  these  are,  in  a 
vacuiim  and  arranged  in  order  of  pitch,  as  follows  : — 


Number  of  Nodal  Lines 

Tone 

Diameters 

Circles 

0 

0 

c 

1 

.       0 

«ba 

2 

0 

/#  +  0-l* 

0 

1 

d'^  +  0-2 

1 

1 

<j'     -0-2 

" 

2 

b'\f  +0-1 

These  tones  rapidly  die  out.  If  the  membranes  sound  in  air,t  or  are  associated 
with  an  air  chamber,  as  in  the  kettledrum,  the  relation  of  the  proper  tones  may 
be  altered.  No  detailed  investigations  have  yet  been  made  on  the  secondary  tones 
of  the  kettledrum.  The  kettledrum  is  used  in  artistic  music,  but  only  to  mark  H 
certain  accents.  It  is  tmied,  indeed,  but  only  to  prevent  injury  to  the  harmony, 
not  for  the  purpose  of  filling  up  chords. 

The  common  character  of  the  instruments  hitherto  described  is,  that,  when 
struck  they  produce  inharmonic  uppei-  partial  tones.  If  these  are  of  nearly  the 
same  pitch  as  the  prime  tone,  their  quality  of  sound  is  in  the  highest  degree  un- 
musical, bad,  and  tinkettly.  If  the  secondary  tones  are  of  very  different  pitcli 
from  the  prime,  and  weak  in  force,  the  quality  of  sound  is  more  musical,  as  for 
example  in  tuning-forks,  harmonicons  of  rods,  and  bells ;  and  such  tones  are  applic- 
able for  marches  and  other  boisterous  music,  principally  intended  to  njark  time. 
But  for  really  artistic  music,  such  instruments  as  these  have  always  been  rejected, 
as  they  ought  to  be,  for  the  inharmonic  secondary  tones,  although  they  rapidly  die 
away,  always  disturb  the  harmony  most  unpleasantly,  renewed  as  they  are  at  every 
fresh  blow.  A  very  striking  example  of  this  was  furnished  by  a  company  of  bell- 
ringers,  said  to  be  Scotch,  that  lately  travelled  about  Germany,  and  performed  alP' 
kinds  of  musical  pieces,  some  of  which  had  an  artistic  character.  The  accuracy 
and  skill  of  the  performance  was  undeniable,  but  the  musical  effect  was  detestable, 
on  account  of  the  heap  of  false  secondary  tones  which  accompanied  the  music, 
although  care  was  taken  to  damp  each  bell  as  soon  as  the  proper  duration  of  its 
note  had  expired,  by  placing  it  on  a  table  covered -with  cloth. 

Sonorous  bodies  with  inharmonic  partials,  may  be  also  set  in  action  by  violin 
bows,  and  then  by  properly  damping  them  in  a  nodal  line  of  the  desired  tone,  the 
secondary  tones  which  lie  near  it  can  be  prevented  from  interfering.  One  simple 
tone  then  predominates  distinctly,  and  it  might  consequently  be  used  for  musical 
purposes.  But  when  the  violin  bow  is  applied  to  any  bodies  with  inharmonic 
upper   partial   tones,    as   tuning-forks,    plates,   bells,   we   hear  a  strong   scratching 

*  [These    decimals    represent   tenths   of  a  numbers  of  vibrations  in  a   second.— 7V«'«i'- 

tone,  or  20  cents  for  the  first  place.     As  there  la  tor.} 

can  be  no  sounds  in  a  vacuum,  these   notes  f  See  ,/.  Boaryet,  L'Institut,  xxxvj 

are   merely   used   to   conveniently   symbolise  pp.  189,  190. 


1870, 


74  MUSICAL  TONES  OF  STRINGS.  part  i. 

sound,  wliich  on  investigation  with  resonators,  is  found  to  consist  mainly  of  these 
same  inharmonic  secondary  tones  of  such  bodies,  not  sounding  continuously  but 
only  in  short  irregular  fits  and  starts.  Intermittent  tones,  as  I  have  already  noted, 
produce  the  effect  of  grating  or  scratching.  It  is  only  when  the  body  excited  by 
the  violin  bow  has  harmonic  upper  partials,  that  it  can  perfectly  accommodate  itself 
to  every  impulse  of  the  bow,  and  give  a  really  musical  quality  of  tone.  The 
reason  of  this  is  that  any  required  periodic  motion  such  as  the  bow  aims  at  pro- 
ducing, can  be  compounded  of  motions  corresponding  to  harmonic  upper  partial 
tones,  but  not  of  other,  inharmonic  vibrations. 

3.  Musical  Tones  of  Strings. 

We  now  proceed  to  the  analysis  of  musical  tones  proper,  which  are  characterised 
H  by  -harmonic  upper  partials.  These  may  be  best  classified  according  to  their  mode 
of  excitement:  1.  By  striking.  2.  By  bowing.  3.  By  blowing  against  a  sharp 
edge.  4.  By  blowing  against  elastic  tongues  or  vibrators.  The  two  first  classes 
comprehend  stringed  instruments  alone,  as  longitudinally  vibrating  rods,  the  only 
other  instruments  producing  harmonic  upper  partial  tones,  are  not  used  for  musical 
purposes.  The  third  class  embraces  flutes  and  the  flute  or  flue  ])ipes  of  organs ; 
the  fourth  all  other  wind  instruments,  including  the  human  voice. 

Strings  excited  bi/  Striking. — Among  musical  instruments  at  present  in  use, 
this  section  embraces  the  pianoforte,  harp,  guitar,  and  zither :  among  physical, 
the  monochord,  arranged  for  an  accurate  examination  of  the  laws  controlling  the 
vibrations  of  strings ;  the  pizzicato  of  bowed  instruments  must  also  be  placed  in 
this  category.  We  have  already  mentioned  that  the  musical  tones  produced  by 
strings  which  are  struck  or  plucked,  contain  numerous  upper  partial  tones.  We 
have  the  advantage  of  possessing  a  complete  theory  for  the  motion  of  plucked 
51  strings,  by  which  the  force  of  their  upper  partial  tones  may  be  determined.  In 
the  last  chapter  we  compared  some  of  the  conclusions  of  this  theory  with  the 
results  of  experiment,  and  found  them  agree.  A  similarly  complete  theory  may  be 
formed  for  the  case  of  a  string  which  has  been  struck  in  one  of  its  points  by  a 
hard  sharp  edge.  The  problem  is  not  so  simple  when  soft  elastic  hammers  are 
used,  such  as  those  of  the  pianoforte,  bnt  even  in  this  case  it  is  possible  to  assign 
a  theory  for  the  motion  of  the  string  which  embraces  at  least  the  most  essential 
features  of  the  process,  and  indicates  the  force  of  the  Tipper  partial  tones.* 

The  force  of  the  upper  partial  tones  in  a  struck  string,  depends  in  general 
on : — 

1 .  The  nature  of  the  stroke. 

2.  The  place  struck. 

3.  The  density,  rigidity,  and  elasticity  of  the  string. 

First,  as  to  the  nature  of  the  stroke.  The  string  may  be  plucked,  by  drawing 
*[1  it  on  one  side  with  the  finger  or  a  point  (the  plectrum,  or  the  ring  of  the  zither- 
player),  and  then  letting  it  go.  This  is  a  us\ial  mode  of  exciting  a  string  in  a  great 
number  of  ancient  and  modern  stringed  instruments.  Among  the  modern,  I  need 
only  mention  the  harp,  guitar,  and  zither.  Or  else  the  string  may  be  struck  with 
a  hammer-shaped  body,  as  in  the  pianoforte.!  I  have  already  remarked  that  the 
strength  and  number  of  the  upper  partial  tones  increases  with  the  number  and 
abruptness  of  the  discontinuities  in  the  motio^i  excited.  This  fact  determines  the 
various  modes  of  exciting  a  string.  When  a  string  is  plucked,  the  finger,  before 
quitting  it,  removes  it  from  its  position  of  rest  throughout  its  whole  length.  A 
discontinuity  in  the  string  arises  only  by  its  forming  a  more  or  less  acute  angle  at 
the  place  where  it  wrajjs  itself  about  the  finger  or  point.  The  angle  is  more  aciite 
for  a  sharp  point  than  for  the  finger.  Hence  the  sharp  point  produces  a  shriller 
tone  with  a  greater  number  of  high  tinkling  upper  partials,  than  the  finger.     But 

*  See  Appendix  V.  be  struck  by  a  hammer-sbaped   body.      See 

t  [I   have   here   omitted   a   few   words   in       pp.  77c  and  l%d'. — Translator.] 
which,  by  an  oversight,  the  spinet  was  said  to 


CHAP.  V.  3.  MUSICAL  TONES  OF  STRINOS.  75 

in  eacli  case  the  iuteiisity  of  the  prime  tone  exceeds  that  of  any  upper  partial.  If 
the  string  is  struck  with  a  sharp-edged  metallic  hammer  which  rebounds  instantly, 
only  the  one  single  point  struck  is  directly  set  in  motion.  Immediately  after  the 
blow  the  remainder  of  the  string  is  at  rest.  It  does  not  move  until  a  wave  of  de- 
flection rises,  and  runs  backwards  and  forwards  over  the  string.  This  limitation 
of  the  original  motion  to  a  single  point  produces  the  most  abrupt  discontinuities, 
and  a  corresponding  long  series  of  upper  partial  tones,  having  intensities,*  in  most 
cases  equalling  or  even  surpassing  that  of  the  prime.  When  the  hammer  is  soft 
and  elastic,  the  motion  has  time  to  spread  before  the  hammer  rebounds.  When 
thus  struck  the  point  of  the  string  in  contact  with  such  a  hammer  is  not  set  in 
motion  with  a  jerk,  but  increases  gradually  and  continuously  in  velocity  during  the 
contact.  The  discontinuity  of  the  motion  is  consequently  much  less,  diminishing 
as  the  softness  of  the  hammer  increases,  and  the  force  of  the  higher  upper  partial 
tones  is  correspondingly  decreased.  II 

We  can  easily  convince  ourselves  of  the  correctness  of  these  statements  by 
opening  the  lid  of  any  pianoforte,  and,  keeping  one  of  the  digitals  down  with  a 
weight,  so  as  to  free  the  string  from  the  damper,  plucking  the  string  at  pleasure 
with  a  finger  or  a  point,  and  striking  it  with  a  metallic  edge  or  the  pianoforte  ham- 
mer itself.  The  qualities  of  tone  thus  obtained  will  be  entirely  diflferent.  When 
the  string  is  struck  or  plucked  with  hard  metal,  the  tone  is  piercing  and  tingling, 
and  a  little  attention  enables  us  to  hear  a  midtitude  of  very  high  partial  tones. 
These  disappear,  and  the  tone  of  the  string  becomes  less  bright,  but  softer,  and 
more  harmonious,  when  we  pluck  the  string  with  the  soft  finger  or  strike  it  with 
the  soft  hammer  of  the  instrument.  We  also  readily  recognise  the  different  loud- 
ness of  the  prime  tone.  When  we  strike  with  metal,  the  prime  tone  is  scarcely 
heard  and  the  quality  of  tone  is  correspondingly  j'oor.  The  peculiar  quality  of 
tone  commonly  termed  poverty,  as  opposed  to  richness,  arises  from  the  upper 
partials  l)eing  comparatively  too  strong  for  the  prime  tone.  The  prime  tone  is  II 
heard  best  when  the  string  is  plucked  with  a  soft  finger,  which  produces  a  rich  and 
yet  harmonious  quality  of  tone.  The  prime  tone  is  not  so  strong,  at  least  in  the 
middle  and  deeper  octaves  of  the  instniment,  when  the  strings  are  struck  with  the 
pianoforte  hammer,  as  when  they  are  plucked  with  the  finger. 

This  is  the  reason  why  it  has  been  found  advantageous  to  cover  pianoforte  ham- 
mers with  thick  layers  of  felt,  rendered  elastic  by  much  compression.  The  outer 
layers  are  the  softest  and  most  yielding,  the  lower  are  firmer.  The  surface  of  the 
hammer  comes  in  contact  with  the  string  without  any  audible  impact ;  the  lower 
layers  give  the  elasticity  which  throws  the  hammer  back  from  the  string.  If  you 
remove  a  pianoforte  hammer  and  strike  it  strongly  on  a  wooden  table  or  against  a 
wall,  it  rebounds  from  them  like  an  india-rubber  ball.  The  heavier  the  hammer 
and  the  thicker  the  layers  of  felt— as  in  the  hammers  for  the  lower  octaves— the 
longer  must  it  be  before  it  rebounds  from  the  string.  The  hammers  for  the  upper 
octaves  are  lighter  and  have  thinner  layers  of  felt.  Clearly  the  makers  of  these  U 
instruments  have  here  been  led  by  practice  to  discover  certain  relations  of  the 
elasticity  (3f  the  hammer  to  the  best  tones  of  the  string.  The  make  of  the  hammer 
has  an  immense  influence  on  the  quality  of  tone.  Theory  shows  that  those  upper 
partial  tones  are  especially  favoured  whose   periodic  time   is   nearly  etpial  to  twice 

*When  iidensitij  is   here  meutioned,  it  is  as   the   pitch  number.      Messrs.   Preece  and 

always  measured  objectively,  by  the  vis  viva  Stroh,  Proc.  IL   S.,  vol.  xxviii.  p.  366,  think 

ov  mechanical  equivalent  of  work  ot  the  corre-  that '  loudness  does  not  depend  upon  ampUtude 

sponding  motion.      [Mr.  Bosanquet  {Acadcmij,  of  vibration  only,  l)ut  upon  the  quantity  of  air 

Dec.  4,  1875,  p.  580,  col.  1)  points  out  that  put  in  vibration ;  and,  therefore,  there  exists 

p.  lOrf,  note,  and  Chap.  IX.,  paragraph  3,  show  an  absolute  physical  magnitude  in  acoustics 

this  measure  to  be  inadmissible,  and  adds  :  analogous  to  that  of  quantity  of  electricity  or 

'  if   we   admit    that   in    similar    organ    pipes  quantity  of   heat,  and  which   may  be  called 

similar  proportions  of  the  wind  supplied  are  quantity  of  sound,'  and  they  illustrate  this  by 

employed  in  the  production  of  tone,  the  me-  the  effect  of   differently  sized  discs   in  then- 

chanical  energy  of   notes    of   given  intensity  automatic  phonograph  there  described.     See 

varies  inversely  as  the  vibration  number,'  i.e.  also  App.  XX.  sect.  M.  No.  2.—Traiis/afo,:\ 


76  MUSICAL  TONES  OF  STRINOS.  part  i. 

the  time  during  which  the  hammer  lies  on  the  string,  and  that,  on  the  other  hand, 
those  disappear  whose  periodic  time  is  6,  10,  14,  etc.,  times  as  great.* 

It  will  generally  be  advantageous,  especially  for  the  deeper  tones,  to  eliminate 
from  the  series  of  upper  partials,  those  which  lie  too  close  to  each  other  to  give  a 
good  compound  tone,  that  is,  from  about  the  seventh  or  eighth  onwards.  Those 
with  higher  ordinal  numbers  are  generally  relatively  weak  of  themselves.  On  ex- 
amining a  new  grand  pianoforte  by  Messrs.  Steinway  of  New  York,  which  was 
remarkable  for  the  evenness  of  its  quality  of  tone,  I  find  that  the  damping  result- 
ing from  the  duration  of  the  stroke  falls,  in  the  deeper  notes,  on  the  ninth  or  tenth 
partials,  whereas  in  the  higher  notes,  the  fourth  and  fifth  partials  were  scarcely  to 
be  got  out  with  the  hammei-,  although  they  were  distinctly  audible  when  the  string 
was  plucked  by  the  nail.+  On  the  other  hand  upon  an  older  and  much  used  grand 
piano,  which  originally  showed  the  principal  damping  in  the  neighbourhood  of  the 

^  seventh  to  the  fifth  partial  for  middle  and  low  notes,  the  ninth  to  the  thirteenth 
partials  are  now  strongly  developed.  This  is  probably  due  to  a  hardening  of  the 
hammers,  and  certainly  can  only  be  prejudicial  to  the  quality  of  tone.  Observa- 
tions on  these  relations  can  be  easily  made  in  the  method  recommended  on  p.  52/v,  c. 
Put  the  point  of  the  finger  gently  on  one  of  the  nodes  of  the  tone  of  which  you 
wish  to  discover  the  strength,  and  then  strike  the  string  by  means  of  the  digital. 
By  moving  the  finger  till  the  required  tone  comes  out  most  purely  and  sounds  the 
longest,  the  exact  position  of  the  node  can  be  easily  found.  The  nodes  which  lie 
near  the  striking  point  of  the  hammer,  are  of  course  chiefly  covei-ed  by  the  damper, 
but  the  corresponding  partials  are,  for  a  reason  to  be  given  presently,  relatively 
weak.  Moreover  the  fifth  partial  speaks  well  when  the  string  is  touched  at  two- 
fifths  of  its  length  from  the  end,  and  the  seventh  at  two-sevenths  of  that  length. 
These  positions  are  of  course  quite  free  of  the  damper.  Generally  we  find  all  the 
partials  which  arise   from  the  method   of  striking  used,  when  we  keep  on  striking 

f  while  the  finger  is  gradually  moved  over  the  length  of  the  string.  Touching  the 
shorter  end  of  the  string  between  the  striking  point  and  the  further  bridge  will  thus 
bring  out  the  higher  partials  from  the  ninth  to  the  sixteenth,  which  are  unisically 
undesirable. 

The  method  of  calculating  the  strength  of  the  individual  upper  partials,  when 
the  diu-ation  of  the  stroke  of  the  hammer  is  given,  will  be  found  further  on. 

Secondly  as  to  the  />Zaw  struck.  In  the  last  chapter,  when  verifying  Ohm's 
law  for  the  analysis  of  musical  tones  by  the  ear,  we  remarked  that  whether  strings 
are  plucked  or  struck,  those  upper  partials  disappear  which  have  a  node  at  the 
point  excited.  Conversely  ;  those  partials  are  comparatively  strongest  which  have 
a  maximum  displacement  at  that  point.  Generally,  when  the  same  method  of 
striking  is  sviccessively  applied  to  different  points  of  a  string,  the  individual  upper 
partials  increase  or  decrease  with  the  intensity  of  motion,  at  the  point  of  excite- 
ment, for  the  coi-responding  simple  vibrations  of  the  string.  The  composition  of 
^  the  musical  tone  of  a  string  can  be  consequently  greatly  varied  by  merely  changing 
the  point  of  excitement. 

Thus  if  a  string  be  struck  in  its  middle,  the  second  partial  tone  disappears, 

*  [The  following   paragraph  on  p.  123  of  several  times.      I   got   out  the   7th   and  9th 

the    1st   English   edition    has   heen   omitted,  harmonic   of  c,    but   on    account   of   difficul- 

and  the  passage  from   '  It  will   generally   be  ties  due  to  the  over-stringing  and  over-barring 

advantageous,'  p.   76(f,  to  '  found  further  on,'  of  the  instrument  and   other   circumstances 

p.  76c,  has  been  inserted,  both  in  accordance  I  did  not  pursue  the  investigation.     Mr.  A.  J. 

with  the  4th  German  edition.—Translator.]  Hipkins  informs  me  that  on  another  occasion 

t  [As   Prof.    Helmholtz   does   not   mention  he   got  out  of  the   c'  string,  struck  at  i  the 

thestrikingdistanceof  the  hammer,  I  obtained  length,    the    Gth,    7th,    8th,    and    9th    har- 

permission  from  Messrs.  Steinway  &  Sons,  at  monies,  as  in  the  experiments  mentioned  in 

their  London  house,  to  examine  the  c,  c'  and  the  next  footnote,  '  the  Gth  and  7th  beautifully 

c"  strings  of  one  of  their  grand  pianos,  and  strong,  the  8tli  and  9th  weaker  but  clear  and 

found  the  striking  distance  to  be  VV.  tV.  ^.nd  unmistakable.'     He  struck  with  the  hammer 

-jY   of   the  length  of   the  string  respectively.  always.     Observe  the  9th  harmonic  of  a  strmg 

I  did  not  measure  the   other  strings,  but  I  struck  with  a  pianoforte  hammer  at  its  node, 

observed   that   the   striking   distances   varied  or  },  its  length. —  Transhi for.'] 


CHAP.  V.   '.]. 


MUSICAL  TONES  OF  STKlNiJS. 


because  it  has  a  node  at  that  point.  But  the  third  partial  tone  comes  out  forcibly, 
because  as  its  nodes  lie  at  I  and  f  the  length  of  the  string  from  its  extremities, 
the  string  is  struck  half-way  between  these  two  nodes.  The  fourth  partial  has  its 
nodes  at  ^>  f  ( =  D'  '^^^^  T  ^^^^  length  of  the  string  from  its  extremity.  It  is  not 
heard,  because  the  point  of  excitement  corresponds  to  its  second  node.  The  sixth, 
eighth,  and  generally  the  partials  with  even  numbers  disappear  in  the  same  way,  but 
the  fifth,  seventh,  ninth,  and  the  other  partials  with  odd  numbers  are  heard.  By 
this  disappearance  of  the  evenly  numbered  partial  tones  when  a  string  is  struck  at  its 
middle,  the  quality  of  its  tone  becomes  peculiar,  and  essentially  different  from  that 
usually  heard  from  strings.  It  sounds  somewhat  hollow  or  nasal.  The  experi- 
ment is  easily  made  on  any  piano  when  it  is  opened  and  the  dampers  are  raised. 
The  middle  of  the  string  is  easily  found  by  trying  where  the  finger  must  be  laid 
to  bring  out  the  first  upper  partial  clearly  and  purely  on  striking  the  digital. 

If  the  string  is  struck  at  i  its  length,  the  third,  sixth,  ninth,  &c.,  partials  U 
vanish.  This  also  gives  a  certain  amount  of  hollowness,  but  less  than  when  the 
string  is  struck  in  its  middle.  When  the  point  of  excitement  approaches  the  end 
of  the  string,  the  prominence  of  the  higher  upper  partials  is  favoured  at  the 
expense  of  the  prime  and  lower  upper  partial  tones,  and  the  sound  of  the  string 
becomes  poor  and  tinkling. 

In  pianofortes,  the  point  struck  is  about  i  to  },  the  length  of  the  string  from 
its  extremity,  for  the  middle  part  of  the  instrument.  We  must  therefore  assume 
that  this  place  has  been  chosen  because  experience  has  shown  it  to  give  the  finest 
musical  tone,  which  is  most  suitable  for  harmonies.  The  selection  is  not  due  to 
theory.  It  results  from  attempts  to  meet  the  requirements  of  artistically  trained 
ears,  and  from  the  technical  experience  of  two  centuries.*     This  gives  particular 


*  [As  my  friend,  Mr.  A.  J.  Hipkius,  of 
Broadwoods',  author  of  a  paper  on  the  '  Historj' 
of  the  Pianoforte,'  in  the  Journal  of  the  Society 
of  Arts  (for  March  9,  1883,  with  additions  on 
Sept.  21,  1883),  has  paid  great  attention  to  the 
archasology  of  the  pianoforte,  and  from  his 
position  at  Messrs.  Broadwoods'  has  the  best 
means  at  his  disposal  for  making  experiments, 
I  requested  him  to  favour  me  with  his  views 
upon  the  subject  of  the  striking  place  and 
harmonics  of  pianoforte  strings,  and  he  has 
obliged  me  with  the  following  observations  : — 
'  Harpsichords  and  spinets,  which  were  set 
in  vibration  by  quill  or  leather  plectra,  had 
no  fixed  point  for  plucking  the  strings.  It 
was  generally  from  about  i  to  i  of  the  vibra- 
ting length,  and  although  it  had  been  observed 
by  Huyghens  and  the  Antwerp  harpsichord- 
maker  Jan  Couchet,  that  a  difference  of  quality 
of  tone  could  be  obtained  by  varying  the 
plucking  place  on  the  same  string,  which  led 
to  the  so-called  lute  stop  of  the  18th  century, 
no  attempt  appears  to  have  been  made  to  gain 
a  uniform  striking  place  throughout  the  scale. 
Thus  in  the  latest  improved  spinet,  a  Hitch- 
cock, of  early  18th  century,  in  my  possession, 
the  striking  place  of  the  c-'s  varies  from  ^  to 
f,  and  in  the  latest  improved  harpsichord,  a 
Kirkman  of  1773,  also  in  my  possession,  the 
striking  distances  vary  from  J  to  ^^  and  for 
the  lute  stop  from  ^  to  ^V  of  the  string,  the 
longest  distances  in  the  bass  of  course,  but 
all  without  apparent  rule  or  proportion.  Nor 
was  any  attempt  to  gain  a  uniform  striking 
place  made  in  the  first  pianofortes.  Stein  of 
Augsburg  (the  favourite  pianoforte-maker  of 
Mozart,  and  of  Beethoven  in  his  virtuoso 
time)  knew  nothing  of  it,  at  least  in  his  early 
instruments.  The  great  length  of  the  bass 
strings  as  carried  out  on  the  single  belly- 
bridge  copied  from  the  harpsichord,  made  a 


reasonable  striking  place  for  that  part  of  the 
scale  impossible. 

'  John  Broadwood,  about  the  year  1788,  was  U 
the  first  to  try  to  equalise  the  scale  in  tension 
and  striking  place.  He  called  in  scientific 
aid,  and  assisted  by  Signor  Cavallo  and  the 
then  Dr.  Gray  of  the  British  Museum,  he 
produced  a  divided  belly-bridge,  which  shorten- 
ing the  too  great  length  of  the  bass  strings, 
permitted  the  establishment  of  a  striking 
place,  which,  in  intention,  should  be  propor- 
tionate to  the  length  of  the  string  throughout. 
He  practically  adopted  a  ninth  of  the  vibrating 
length  of  the  string  for  his  striking  place, 
allowing  some  latitude  in  the  treble.  This 
division  of  the  belly-bridge  became  universally 
adopted,  and  with  it  an  approximately  rational 
striking  place. 

'  Carl  Kiitzing  (Das  Wissenschaftliehe  der 
Fortcpiano-Baukunst,  1844,  p.  41)  was  enabled 
to  propomid  from  experience,  that  J  of  the 
length  of  the  string  was  the  most  suitable 
distance  in  a  pianoforte  for  obtaining  the  best  ^ 
quality  of  tone  from  the  strings.  The  love  of 
noise  or  effect  has,  however,  inclined  makers  to 
shorten  distances,  particularly  in  the  trebles. 
Kiitzing  appears  to  have  met  with  I  in  some 
instances,  and  Helmholtz  has  adopted  that 
very  exceptional  measure  for  his  table  on 
p.  79f.  I  cannot  say  I  have  ever  met  with  a 
striking  place  of  this  long  distance  from  the 
wrestplauk-bridge.  The  present  head  of  the 
firm  of  Broadwood  (Mr.  Henry  Fowler  Broad- 
wood)  has  arrived  at  the  same  conclusions  as 
Kiitzing  with  respect  to  the  superiority  of  the 
^th  distance,  and  has  introduced  it  in  his 
pianofortes.  At  Jth  the  hammers  have  to  be 
softer  to  get  a  like  quality  of  tone ;  an  equal 
system  of  tension  being  presupposed. 

'  According  to  Young's  law,  which  Helm- 
holtz by  experiment  confirms,  the  impact  of 


78  MUSICAL  TONES  OF  STRINGS.  part  i. 

interest  to  the  investigation  of  the  composition  of  nnisicul  tones  for  this  point  of 
excitement.  An  essential  advantage  in  the  choice  of  this  position  seems  to  be 
that  the  seventh  and  ninth  partial  tones  disappear  or  at  least  become  very  weak. 
These  are  the  first  in  the  series  of  partial  tones  which  do  not  belong  to  the  major 
chord  of  the  prime  tone.  Up  to  the  sixth  partial  we  have  only  Octaves,  Fifths, 
and  major  Thirds  of  the  prime  tone  ;  the  seventh  is  nearly  a  minor  Seventh,  the 
ninth  a  major  Second  of  the  prime.  Hence  these  will  not  fit  into  the  major 
chord.  Experiments  on  pianofortes  show  that  when  the  string  is  struck  by  the 
hammer  and  touched  at  its  nodes,  it  is  easy  to  bring  out  the  six  first  partial  tones 
(at  least  on  the  strings  of  the  middle  and  lower  octaves),  but  that  it  is  either  not 
possible  to  bring  out  the  seventh,  eighth,  and  ninth  at  all,  or  that  we  obtain  at 
best  very  weak  and  imperfect  resvilts.  The  difficulty  here  is  not  occasioned  by  the 
incapacity  of  the  string  to  form  such  short  vibrating  sections,  for  if  instead  of  striking 

H  the  digital  we  pluck  the  string  nearer  to  its  end,  and  damp  the  corresponding 
nodes,  the  seventh,  eighth,  ninth,  nay  even  the  tenth  and  eleventh  partial  may  be 
clearly  and  brightly  produced.  It  is  only  in  the  upper  octaves  that  the  strings  are 
too  short  and  stiff  to  form  the  high  upper  partial  tones.  For  these,  several  instru- 
ment-makers place  the  striking  point  nearer  to  the  extremity,  and  thus  obtain  a 
brighter  and  more  penetrating  tone.  The  upper  partials  of  these  strings,  which 
their  stiffness  renders  it  difficidt  to  bring  out,  are  thus  favoured  as  against  the 
prime  tone.  A  similarly  brighter  tone,  but  at  the  same  time  a  thinner  and  poorer 
one,  can  be  obtained  from  the  lower  strings  by  placing  a  bridge  nearer  the  striking 
point,  so  that  the  hammer  falls  at  a  point  less  than  i  of  the  effective  length  of  the 
string  from  its  extremity. 

While  on  the  one  hand  the  tone  can  be  rendered  more  tinkling,  shrill,  and 
acute,  by  striking  the  string  with  hard  bodies,  on  the  other  hand  it  can  be  rendered 
duller,  that  is,  the  prime  tone  may  be  made  to  outweigh  the  upper  partials,  by 

H  striking  it  with  a  soft  and  heavy  hammer,  as,  for  example,  a  little  iron  hammer 
covered  with  a  thick  sheet  of  india-rubber.  The  strings  of  the  lower  octaves  then 
produce  a  much  fuller  but  duller  tone.  To  compare  the  different  qualities  of  tone 
thus  produced  by  using  hammers  of  different  constructions,  care  must  be  taken 
always  to  strike  the  string  at  the  same  distance  from  the  end  as  it  is  struck  by  the 
proper  hammer  of  the  instrument,  as  otherwise  the  results  would  be  mixed  up  with 
the  changes  of  quality  depending  on  altering  the  striking  point.  These  circum- 
stances  are  of  course  well  known  to   the   instrument-makers,   because  they  have 

the  hammer  abolishes  the  node  of  the  striking  diately  after  production,  they  last  much  longer 

place,  and  with  it  the  partial  belonging  to  it  and  are  much  brighter. 

throughout  the  string.     I  do  not  find,  however,  '  I  do  not  think  the  treble  strings  are  from 

that  the  hammer  striking  at  the  jth  elimi-  shortness  and  stiffness  incapable  of  forming 

nates  the  8th  partial.     It  is  as  audible,  when  high  proper  tones.      If  it  were  so  the  notes 

touched  as  an  harmonic,  as  the  9th  and  higher  would  be  of  a  very  different  quality  of  tone  to 

partials.     It  is  easy,  on  a  long  string  of  say  that  which  they  are  found  to  have.     Owing  to 

51  from  25  to  45  inches,  to  obtain  the  series  of  the  very  acute  pitch  of  these  tones  our  ears 

upper   partials   up   to   the   fifteenth.      On    a  cannot   follow   them,   but   their   existence   is 

string  of  45  inches  I  have  obtained  as  far  as  proved   by  the   fact   that  instrument-makers 

the  23rd  harmonic,  the  diameter  of  the  wire  often   bring   their   treble  striking  place  very 

being  1-17  mm.  or  -07  inches,  and  the  tension  near  the  wrestplank-bridge  in  order  to  secure 

being   71    kilogrammes    or    156-6    lbs.       The  a  brilliant  tone  effect,  or  ring,   by  the  pre- 

partials   diminish   in   intensity  with   the  re-  ponderance  of  these  harmonics, 

duction  of  the  vibrating  length;    the  2nd  is  'The     clavichord     differs     entirely     from 

stronger  than  the  3rd,  and  the  3rd  than  the  hammer  and  plectrum  keyboard  instruments 

4th,  &c.     Up  to  the  7tli  a  good  harmonic  note  in  the  note  being  started  from  the  end,  the 

can  always  be  brought  out.     After  the  8th,  as  tangent    (brass   pin)  which   stops   the  string 

Helmholtz  says,   the  higher  partials  are  all  being  also  the  means  of  exciting  the  sound, 

comparatively  weak   and    become    gradually  But  the  thin  brass  wires  readily  break  up  into 

fainter.     To  strengthen  them  we  may  use  a  segments  of  short  recurrence,  the  bass  wires, 

narrower   harder  hammer.      To    hear    them  which  are  most  indistinct,  being  helped  in  the 

with  an  ordinary  hammer  it  is  necessary  to  latest  instruments  by  lighter  octave  strings, 

excite  them  by  a  firm  blow  of  the  hand  upon  which  serve  to  make  the  fundamental  tones 

the  finger-key  and  to  continue  to  hold  it  down.  apparent.'     See  also  the  last  note,  p.  76d',  and 

They  sing  out  quite  clearly  and  last  a  very  App.  XX.  sect.  N. —  Translator.] 
sensible  time.     On  removing  the  stop  imme- 


CHAP.  V.   3. 


MUSICAL  TONES   OF   STRINGS. 


79 


themselves  selected  heavier  and  softer  hammers  for  the  lower,  and  lighter  and 
harder  for  the  upper  octaves.  But  when  we  see  that  they  have  not  given  more 
than  a  certain  weight  to  the  hammers  and  have  not  increased  it  sufficiently  to 
reduce  the  intensity  of  the  upper  partial  tones  still  further,  we  feel  convinced  that 
a  musically  trained  ear  prefers  that  an  instrument  to  be  used  for  rich  combinations 
of  harmony  should  possess  a  quality  of  tone  which  contains  upper  partials  with  a 
certain  amount  of  strength.  In  this  respect  the  composition  of  the  tones  of 
pianoforte  strings  is  of  great  interest  for  the  whole  theory  of  music.  In  no  other 
instrument  is  there  so  wide  a  field  for  alteration  of  quality  of  tone  ;  in  no  other, 
then,  was  a  musical  ear  so  unfettered  in  the  choice  of  a  tone  that  would  meet  its 
wishes. 

As  I  have  already  observed,  the  middle  and  lower  octaves  of  pianoforte  strings 
generally  allow  the  six  first  partial  tones  to  be  clearly  produced  by  striking  the 
digital,  and  the  three  first  of  them  are  strong,  the  fifth  and  sixth  distinct,  but  much  K 
weaker.  The  seventh,  eighth,  and  ninth  are  eliminated  by  the  position  of  the 
striking  point.  Those  higher  than  the  ninth  are  always  very  weak.  For  closer 
comparison  I  subjoin  a  table  in  which  the  intensities  of  the  partial  tones  of  a  string 
for  different  methods  of  striking  have  been  theoretically  calcidated  from  the 
formulfe  developed  in  the  Appendix  V.  The  effect  of  the  stroke  of  a  hammer 
depends  on  the  length  of  time  for  which  it  touches  the  string.  This  time  is  given 
in  the  table  in  fractions  of  the  periodic  time  of  the  prime  tone.  To  this  is  added 
a  calculation  for  strings  plucked  by  the  finger.  The  striking  point  is  always 
assumed  to  be  at  i  of  the  length  of  the  string  from  its  extremity. 

Theoretical  Intensity  of  the  Partial  Tones  of  Strings. 


striking  point  at  f  of  tlie  length  of  the  string 


I    Number  of 

[    the  Partial 

Tone 


Excited  by 
Plucking 


100 
81-2 
56-1 
31-6 
13 

2-8 

0 


Struck  by  a  hammer  which  touches  the  string  for 

f  I  fV  I  t\.         I 

of  the  periodic  time  of  the  prime  tone 


100 
99-7 
8-9 
2-3 
1-2 
001 
0 


100 
189-4 
107-9 
17-3 

0 

0-5 

0 


100 
249 
242-9 
118-9 
26-1 

1-3 

0 


100 

285-7 
357-0 
259-8 
108-4 
18-8 
0 


Struck  by  a 

perfect  hard 

Hammer 


100 
324-7 
504-9 
504-9 
324-7 
100-0 
0 


For  easier  comparison  the  intensity  of  the  prime  tone  has  been  throughout 
assumed  as  100.  I  have  compared  the  calculated  intensity  of  the  upper  partials 
with  their  force  on  the  grand  pianoforte  already  mentioned,  and  found  that  the 
first  series,  under  f,  suits  for  about  the  neighbourhood  of  c".  In  higher  parts  of  U 
the  instrument  the  upper  partials  were  much  weaker  than  in  this  colunm.  On 
striking-  the  digital  for  c",  I  obtained  a  powerful  second  partial  and  an  almost  in- 
audible third.  The  second  column,  marked  y'^,  corresponded  nearly  to  the  region  of 
[I,  the  second  and  third  partials  were  very  strong,  the  fourth  partial  was  weak. 
The  third  column,  inscribed  f ^,  corresponds  with  the  deeper  tones  from  c'  down- 
wards ;  here  the  four  first  partials  are  strong,  and  the  fifth  weaker.  In  the  next 
column,  under  T.'ij,  the  third  partial  tone  is  stronger  than  the  second  ;  there  was 
no  corresponding  note  on  the  pianoforte  which  I  examined.  With  a  perfectly  hard 
hammer  the  third  and  fourth  partials  have  the  same  strength,  and  are  stronger 
than  all  the  others.  It  results  from  the  calculations  in  the  above  table  that  piano- 
forte tones  in  the  middle  and  lower  octaves  have  their  fundamental  tone  weaker 
than  the  first,  or  even  than  the  two  first  upper  partials.  This  can  also  be  con- 
firmed by  a  comparison  with  the  effects  of  plucked  strings.  For  the  latter  the 
second  partial   is  weaker  than   the   first ;    and   it   will   be   found    that   the  prime 


80  MUSICAL  TONES  OF  BOWED  INSTRUMENTS.  part  i. 

tone  is  much  more  distinct  in  the  tones  of  pianoforte  strings  when  phicked  by  the 
finger,  than  when  struck  by  the  hammer. 

Although,  as  is  shown  by  the  mechanism  of  the  upper  octaves  on  pianofortes, 
it  is  possible  to  produce  a  compound  tone  in  which  the  prime  is  predominant, 
makers  have  preferred  arranging  the  luethod  of  striking  the  lower  strings  in  such 
a  way  as  to  preserve  the  five  or  six  first  partials  distinctly,  and  to  give  the  second 
and  third  greater  intensity  than  the  prime. 

Thirdly,  as  regards  the  thickness  and  material  of  the  strings.  Very  rigid 
strings  will  not  form  any  very  high  upper  partials,  because  they  carmot  readily 
assume  inflections  in  opposite  directions  within  very  short  sections.  This  is  easily 
observed  by  stretching  two  strings  of  different  thicknesses  on  a  monochord  and 
endeavouring  to  produce  their  high  upper  partial  tones.  We  always  succeed  much 
better  with  the  thinner  than  with  the  thicker  string.  To  produce  very  high  upper 
^  partial  tones,  it  is  preferable  to  use  strings  of  extremely  fine  wire,  such  as  gold  lace 
makers  employ,  and  when  they  are  excited  in  a  suitable  manner,  as  for  example  by 
plucking  or  striking  with  a  metal  point,  these  high  iipper  partials  may  be  heard  in 
the  compound  itself.  The  numerous  high  upper  partials  which  lie  close  to  each 
other  in  the  scale,  give  that  peculiar  high  inharmonious  noise  which  we  are 
accustomed  to  call  '  tinkling '.  From  the  eighth  partial  tone  upwards  these  simple 
tones  are  less  than  a  whole  Tone  apart,  and  from  the  fifteenth  upwards  less  than  a 
Semitone.  They  consequently  form  a  series  of  dissonant  tones.  On  a  string  of 
the  finest  iron  wire,  such  as  is  used  in  the  manufacture  of  artificial  flowers,  700 
centimetres  (22'97  feet)  long,  I  was  able  to  isolate  the  eighteenth  partial  tone.  The 
peculiarity  of  the  tones  of  the  zither  depends  on  the  presence  of  these  tinkling 
upper  partials,  but  the  series  does  not  extend  so  far  as  that  just  mentioned,  because 
the  strings  are  shorter. 

Strings  of  gut  are  much  lighter  than  metal  strings  of  the  same  compactness, 
^  and  hence  produce  higher  partial  tones.  The  difference  of  their  musical  quality 
depends  partly  on  this  circumstance  and  partly  on  the  inferior  elasticity  of  the  gut, 
which  damps  their  partials,  especially  their  higher  partials,  much  more  rapidly. 
The  tone  of  plucked  cat-gut  strings  {guitar,  harp)  is  consequently  much  less 
tinkling  than  that  of  metal  strings. 

4.  Miisical  Tones  of  Bowed  Instruments. 

No  complete  mechanical  theory  can  yet  be  given  for  the  motion  of  strings 
excited  by  the  violin-bow,  because  the  mode  in  which  the  bow  affects  the  motion 
of  the  string  is  unknown.  But  by  applying  a  peculiar  method  of  observation, 
proposed  in  its  essential  features  by  the  French  physicist  Lissajous,  I  have  found 
it  possible  to  observe  the  vibrational  form  of  individual  points  in  a  violin  string, 
and  from  this  observed  form,  which  is  comparatively  very  simple,  to  calculate  the 
H  whole  motion  of  the  string  and  the  intensity  of  the  vipj)er  partial  tones. 

Look  through  a  hand  magnifying  glass  consisting  of  a  strong  convex  lens,  at 
any  small  bright  object,  as  a  grain  of  starch  reflecting  a  flame,  and  appearing  as  a 
fine  point  of  light.  Move  the  lens  about  while  the  point  of  light  remains  at  rest, 
and  the  point  itself  will  appear  to  move.  In  the  apparatus  I  have  employed,  which 
is  shown  in  fig.  22  opposite,  this  lens  is  fastened  to  the  end  of  one  prong  of  the 
tuning-fork  0,  and  marked  L.  It  is  in  fact  a  combination  of  two  achromatic 
lenses,  like  those  used  for  the  object-glasses  of  microscopes.  These  two  lenses 
may  be  used  alone  as  a  doublet,  or  be  combined  with  others.  When  more 
magnifying  power  is  required,  we  can  introduce  behind  the  metal  plate  A  A,  which 
carries  the  fork,  the  tube  and  eye-piece  of  a  microscope,  of  which  the  doublet  then 
forms  the  object-glass.  This  instrument  may  he  called  a  vibration  microscojie. 
AVhen  it  is  so  arranged  that  a  fixed  luminous  point  may  be  clearly  seen  through  it, 
and  the  fork  is  set  in  vibration,  the  doublet  L  moves  periodically  up  and  down  in 
pendular  vibration.     The  observer,   however,   appears   to   see   the   luminous  point 


CHAP.  V 


4.  MUSICAL  TONES  OF   BOWED  TNSTIUMENT 


SI 


itself  vibrate,  and,  since  the  separate  vibrations  succeed  each  otlier  so  rapidly  that 
the  impression  on  the  eye  cannot  die  away  during-  the  time  of  a  whole  vibration, 
the  path  of  the  luminous  point  appears  as  a  fixei  straight  line,  increasing  in  length 
with  the  excursions  of  the  fork.'"' 

The  grain  of  starch  which  reflects  tlie  light  to  be  seen,  is  then  fastened  to  the 
resonant  "body  whose  vibrations  we  intend  to  observe,  in  such  a  way  that  the  grain 
moves  backwards  and  forwards  horizontally,  while  the  doublet  moves  up  and  down 
vertically.  AVhen  both  motions  take  place  at  once,  the  observer  sees  the  real 
horizontal  motion  of  the  luminous  point  combined  with  its  apparent  vertical  motion, 
and  the  combination  results  in  an  apparent  curvilinear  motion.  The  field  of  vision 
in    tlie    nucroscope    then    shows    an    apparently   steady    and    unchangeable    bright 


curve,  when  either  the  periodic  times  of  the  vibrations  of  the  grain  of  starch  and  11 
of  the  tuning-fork  are  exactly  equal,  or  one  is  exactly  two  or  three  or  four  times  as 
great  as  the  other,  because  in  this  case  the  luminous  point  passes  over  exactly  the 
same  path  every  one  or  every  two,  three,  or  four  vibrations.  If  these  ratios  of  the 
vibrational  numbers  are  not  exactly  perfect,  the  curves  alter  slowly,  and  the  effect 
to  the  eye  is  as  if  they  were  drawn  on  the  surface  of  a  transparent  cylinder  which 
slowly  revolved  on  its  axis.  This  slow  displacement  of  the  apparent  cui^-es  is  not 
disadvantageous,  as  it  allows  the  observer  to  see  them  in  different  positions.  But 
if  the  ratio  of  the  pitch  numbers  of  the  observed  body  and  of  the  fork  differs  too 


*  The  end  of  the  other  prong  of  the  fork 
is  thickened  to  counterbalance  the  weight  of 
the  doublet.  The  iron  loop  B  which  is  clamped 
on  to  one  prong  serves  to  alter  the  pitch  of 
the  fork  slightly  ;  we  flatten  the  pitch  by 
moving  tbe  loop  towards  tlie  end  of  the  prong. 


E  is  an  electro-magnet  by  which  the  fork  is 
kept  in  constant  uniform  vibration  on  passing 
intermittent  electrical  currents  through  its 
wire  coils,  as  will  be  described  more  in  detail 
in  Chapter  VI. 


82 


MUSICAL  TONES  OF  BOWED  INSTRUMENTS. 


much  from  one  expressible  by  small  whole  numbers,  the  motion  of  the  curve  is  too 
rapid  for  the  eye  to  follow  it,  and  all  becomes  confusion. 

If  the  vibration  microscope  has  to  be  used  for  observing  the  motion  of  a  violin 
string,  the  luminous  point  must  be  attached  to  that  string.  This  is  done  by  first 
blackening  the  reqiiired  spot  on  the  string  with  ink,  and  when  it  is  dry,  rubbing  it 
over  with  wax,  and  powdering  this  with  starch  so  that  a  few  grains  remain  sticking. 
The  violin  is  then  fixed  with  its  strings  in  a  vertical  direction  opposite  the  micro- 
scope, so  that  the  luminous  reflection  from  one  of  the  grains  of  starch  can  be 
clearly  seen.  The  bow  is  drawn  across  the  strings  in  a  direction  parallel  to  the 
prongs  of  the  fork.  Every  point  in  the  string  then  moves  horizontally,  and  on 
setting  the  fork  in  motion  at  the  same  time,  the  observer  sees  the  peculiar 
vibrational  curves  already  mentioned.  For  the  purposes  of  observation  I  used  the 
a  string,  which  I  tuned  a  little  higher,  as  //[?,  so  that  it  was  exactly  two  Octaves 
H  higher  than  the  tuning-fork  of  the  microscope,  which  sounded  B\}. 

In  fig.  23  are  shown  the  resulting  vibrational  curves  as  seen  in  the  vibration 
microscope.      The  straight  horizontal  lines  in  the  figures,  a  to  a,  b  to  It,  c  to  c 


show  the  apparent  path  of  the  observed  luminous  point,  before  it  had  itself  been 
set  in  vibration ;  the  curves  and  zigzags  in  the  same  figures,  show  the  a^^parent 
path  of  the  luminous  point  when  it  also  was  made  to  move.  By  their  side,  in  A, 
B,  C,  the  same  vibrational  forms  are  exhibited  according  to  the  methods  used  in 
Chapters  I.  and  II.,  the  lengths  of  the  horizontal  line  being  directly  proportional 
to  the  corresponding  lengths  of  time,  whereas  in  figures  a  to  a,  b  to  b,  c  to  c,  the 
horizontal  lengths  are  proportional  to  the  excursions  of  the  vibrating  microscope. 
A,  and  a  to  a,  show  the  vibrational  curves  for  a  tuning-fork,  that  is  for  a  simple 
pendular  vibration  ;  B  and  b  to  b  those  of  the  middle  of  a  violin  string  in  unison 
with  the  fork  of  the  vibration  microscope ;  C  and  c,  c,  those  for  a  string  which  was 
tuned  an  Octave  higher.  We  may  imagine  the  figures  a  to  a,  b  to  b,  and  c  to  c,  to 
be  formed  from  the  figures  A,  B,  C,  by  supposing  the  surface  on  which  these  are 
drawn  to  be  wrapped  round  a  transparent  cylinder  whose  circumference  is  of  the 
same  length  as  the  horizontal  line.  The  curve  di-awn  upon  the  surface  of  the 
cylinder  must  then  be  observed  from  such  a  point,  that  the  horizontal  line  which 
when  wrapped  round  the  cylinder  forms  a  circle,  appears  perspectively  as  a  single 
straight  line.  The  vibrational  curve  A  will  then  appear  in  the  forms  a  to  a,  B  in 
the  forms  b  to  b,  C  in  the  forms  c  to  c.  When  the  pitch  of  the  two  vibrating 
bodies  is  not  in  an  exact  harmonic  ratio,  this  imaginary  cylinder  on  which  the 
vibrational  curves  are  drawn,  appears  to  revolve  so  that  the  forms  a  to  a,  &c.,  are 
assumed  in  succession. 

It  is  now  easy  to  rediscover  the  forms  A,  B,  C,  from  the  forms  a  to  a,  b  to  b, 


CHAP.  V.  4.  MUSICAL  TONES  OF  BOWED  INSTllUMENTS.  83 

and  c  to  c,  and  as  the  fonner  give  a  more  intelligible  image  of  the  motion  of  the 
string  than  the  latter,  the  curves,  which  are  seen  as  if  they  were  traced  on  the 
surface  of  a  cylinder,  will  be  drawn  as  if  their  trace  had  been  unrolled  from  the 
cylinder  into  a  plane  figure  like  A,  B,  C.  The  meaning  of  our  vibrational  curves 
will  then  precisely  correspond  to  the  similar  curves  in  preceding  chapters.  When 
four  vibrations  of  the  violin  string  correspond  to  one  vibration  of  the  fork  (as  in 
our  experiments,  where  the  fork  gave  £\f  and  the  string  f/\},  p.  82a),  so  that 
four  waves  seem  to  be  traced  on  the  surface  of  the  imaginary  cylinder,  and  when 
moreover  they  are  made  to  rotate  slowly  and  are  thus  viewed  in  diflferent  positions, 
it  is  not  at  all  difficult  to  draw  them  from  immediate  inspection  as  if  they  had 
been  rolled  off  on  to  a  plane,  for  the  middle  jags  have  then  nearly  the  same 
appearance  on  the  cylinder  as  if  they  were  traced  on  a  plane. 

The  figures  23  B  and  23  C  (p.  82/>),  immediately  give  the  vibrational  forms  for 
the  middle  of  a  violin  string,  when  the  bow  bites  well,  and  the  prime  tone  of  the  H 
string  is  fully  and  powerfully  produced.  It  is  easily  seen  that  these  vibrational 
forms  are  essentially  different  from  that  of  a  simple  vibration  (fig.  23,  A).  When 
the  point  is  taken  nearer  the  ends  of  the  string  the  vibrational  figure  is  shown  in 
fig.  24,  A,  and  the  two  sections  a^,  /8y,  of  any  wave,  are  to  one  another  as  the  two 
sections  of  the  string  which  lie  on  either  side  of  the  observed  point.     In  the  figure 


this  ratio  is  3  :  1 ,  the  point  being  at  i  the  length  of  the  string  from  its  extremity. 
Close  to  the  end  of  the  string  the  form  is  as  in  fig.   24,  B.    -The  short  lengths  of 
line  in  the  figure  have  been  made  faint  because  the  corresponding  motion  of  the  ^ 
luminous  point  is  so  rapid  that  they  often  become  invisible,  and  the  thicker  lengths 
are  alone  seen.* 

These  figures  show  that  every  point  of  the  string  Ijetween  its  two  extremities 
vibrates  with  a  constant  velocity.  For  the  middle  point,  the  velocity  of  ascent  is 
equal  to  that  of  descent.  If  the  violin  bow  is  used  near  the  right  end  of  the 
string  descending,  the  velocity  of  descent  on  the  right  half  of  the  string  is  less 
than  that  of  ascent,  and  the  more  so  the  nearer  to  the  end.  On  the  left  half  of 
the  string  the  converse  takes  place.  At  the  place  of  bowing  the  velocity  of  descent 
appears  to  be  equal  to  that  of  the  violin  bow.  During  the  greater  part  of  each 
vibration  the  string  here  clings  to  the  bow,  and  is  carried  on  by  it;  then  it  suddenly 
detaches  itself  and  rebounds,  whereupon  it  s  seized  by  other  points  in  the  bow  and 
again  carried  forward.f 

Our  present  purpose  is  chiefly  to  determine  the  upper  partial  tones.  The 
vibrational  forms  of  the  individual  points  of  the  string  being  known,  the  intensity  ^ 
of  each  of  the  partial  tones  can  be  completely  calculated.  The  necessary  mathe- 
matical formulre  are  developed  in  Appendix  VI.  The  following  is  the  result  of  the 
calculation.  When  a  string  excited  by  a  violin  bow  speaks  well,  all  the  upper 
partial  tones  which  can  be  formed  by  a  string  of  its  degree  of  rigidity,  are  present, 
and  their  intensity  diminishes  as  their  pitch  increases.  The  amplitude  and  the 
intensity  of  the  second  partial  is  one-fourth  of  that  of  the  prime  tone,  that  of  the 

*  [Dr.  Huggiiis,  F.R.S.,  on  experimenting,  string  has  been  given  by  Herr  Clem.  Neumann 

finds   it   probable   that   under   the   bow,    the  in    the    Frocecdrnf/s   {Sitmnysberkhte)    of    the 

relative   velocity   of   descent   to   that   of   the  /.  and  E.  Academy  at  Vienna,  mathematical 

rebound  of  the  string,  or  ascent,  is  influenced  and  physical  class,  vol.  Ixi.  p.  89.    He  fastened 

by   the   tension    of   the  hairs  of   the  bow. —  bits  of  wire  in  the  form  of  ar  comb  to  the  bow 

Translator.]  itself.     On   looking  through    this   grating  at 

t  These    facts  suffice    to    determine     the  the   string    the   observer    sees   a    system    of 

complete   motion     of    bowed     strings.       See  rectilinear  zigzag  lines.     The  conclusions  as 

Appendix  VI.     A   much   simpler   method  of  to  the  mode  of  motion  of   the  string   agree 

observing    the   vibrational    form   of  a   violin  with  those  given  above. 

G    2 


84  MUSICAL  TONES  OF  BOWED  INSTRUMENTS.  i>art  i. 

third  partial  a  ninth,  that  of  the  fourth  a  sixteenth,  and  so  on.  This  is  the  same 
scale  of  intensity  as  for  the  partial  tones  of  a  string  plncked  in  its  middle,  with 
this  exception,  that  in  the  latter  case  the  evenly  numbered  partials  all  disappear, 
whei'eas  they  are  all  present  when  the  how  is  iised.  The  upper  partials  in  the 
compound  tone  of  a  violin  are  heard  easily  and  wall  be  found  to  be  strong  in  sound 
if  they  have  been  first  produced  as  so-called  harmonics  on  the  string,  by  bowing 
lightly  while  gently  touching  a  node  of  the  required  partial  tone.  The  strings  of 
a  violin  will  allow  the  harmonics  to  be  produced  as  high  as  the  sixth  partial  tone 
with  ease,  and  with  some  difficulty  even  up  to  the  tenth.  The  lower  tones  speak 
best  when  the  string  is  bowed  at  from  one-tenth  to  one-twelfth  the  length  of  the 
vibrating  portion  of  the  string  from  its  extremity.  P'or  the  higher  harmonics 
where  the  sections  are  smaller,  the  strings  must  be  bowed  at  about  one-fourth  or 
one-sixth  of  their  vibi'ating  length  from  the  end.* 

U  The  prime  in  the  compound  tones  of  bowed  instruments  is  comparatively  more 
powerful  than  in  those  produced  on  a  pianoforte  or  guitar  by  striking  or  plucking 
the  strings  near  to  their  extremities ;  the  first  upper  partials  are  comparatively 
weaker ;  but  the  higher  upper  partials  from  the  sixth  to  about  the  tenth  are  much 
more  distinct,  and  give  these  tones  their  cutting  character. 

The  fundamental  form  of  the  vibrations  of  a  violin  string  just  described,  is, 
when  the  string  speaks  well,  tolerably  independent  of  the  place  of  bowing,  at  least 
in  all  essential  features.  It  does  not  in  any  respect  alter,  like  the  vibrational  form 
of  struck  or  plucked  strings,  according  to  the  position  of  the  point  excited.  Yet 
there    are    certain     obser-  Fk;.  25. 

vable  differences  of  the 
vibrational  figure  which 
depend  upon  the  bowing- 
point.     Little  crumples  are 

H  usually  perceived  on  the 
lines  of  the  vibrational 
figure,  as  in  fig.  25,  which 
increase  in  breadth  and  height  the  further  the  bow  is  removed  from  the  extremity 
of  the  string.  When  we  bow  at  a  node  of  one  of  the  higher  upper  partials 
which  is  near  the  bridge,  these  crumples  are  simply  reduced  by  the  absence  of 
that  part  of  the  normal  motion  of  the  string  which  depends  on  the  partial  tones 
having  a  node  at  that  place.  When  the  observation  on  the  vibrational  form  is 
made  at  one  of  the  other  nodes  belonging  to  the  deepest  tone  which  is  elimi- 
nated, none  of  these  crumples  are  seen.  Thus  if  the  string  is  bowed  at  4th, 
or  ~ths,  or  |ths,  or  iths,  &c.,  of  its  length  from  the  bridge,  the  vibrational 
figure  is  simple,  as  in  fig.  24  (p.  83/y).  But  if  we  observe  between  two  nodes, 
the  crumples  appear  as  in  fig.  25.  Variations  in  the  quality  of  tone  partly 
depend    upon   this    condition.      When    the    violin    bow    is   brought    too    near    the 

II  finger  board,  the  end  of  which  is  ith  the  length  of  the  string  from  the  bridge, 
the  5th  or  6th  partial  tone,  which  is  generally  distinctly  audible,  will  be  absent. 
The  tone  is  thus  rendered  duller.  The  usual  place  of  bowing  is  at  about  y\jth 
of  the  length  of  the  string ;  for  p^no  passages  it  is  somewhat  further  from 
the  bridge  and  for  forte  somewhat  nearer  to  it.  If  the  bow  is  brought  near  the 
bridge,  and  at  the  same  time  but  lightly  pressed,  another  alteration  of  quality 
occurs,  which  is  readily  seen  on   the  vibrational  figure.     A  mixture   is  formed  of 

*  [The  position  of  the  finger  for  producing  near   the    nut,    out   of   165   mm.   the   actual 

the  harmonic  is  often  sHghtly  different  from  half  length  of  the  strings.     These  differences 

that   theoretically   assigned.       Dr.    Huggins,  must    therefore    be   due   to    some    imperfec- 

F.R.S.,    kindly  tried  for  me  the  position  of  tions  of  the  strings  themselves.     Dr.  Huggins 

the   Octave  harmonic  on  the  four  strings  of  finds   that  there  is  a  space  of  a  quarter  of 

his    Stradivari,    a   mark  with   Chinese   white  an  inch   at   any  point  of  which  the  Octave 

being  made  under  his  finger   on   the  finger  harmonic  may  be  brought  out,  but  the  quality 

board.      Result,    1st    and    4th   string   exact,  of  tone  is  best  at  the  points  named  above. — 

2nd  string  3  mm.,  and  3rd  string  5  mm.  too  Trmislator.] 


CHAP.  V.  4.  MUSICAL  TONES  OF  BOWED  INSTIU'.MEXTS.  85 

the  prime  tone  and  first  liarmonic  of  the  string.  By  liglit  and  rapid  howing, 
namely  at  about  oVth  of  the  length  of  the  string  from  the  bridge,  wc  sometimes 
obtain  the  upper  Octave  of  tlie  prime  tone  by  itself,  a  node  being  formed  in  the 
middle  of  the  string.  On  bowing  more  firmly  the  prime  tone  innnediately  sounds. 
Intermediately  the  higher  Octave  may  mix  with  it  in  any  proportion.  This  is 
immediately  recognised  in  the  vibrational  figure.  Fig.  26  gives  the  corresponding 
series  of  forms.  It  is  seen  how  a  fresh  crest  appears  on  the  longer  side  of  the 
front  of  a  wave,  jutting  out  at  first  slightly,  then  more  strongly,  till  at  length  the 
crests  of  the  new  waves  are  as  high  as  those  of  the  old,  and  then  the  vibrational 
number  has  doubled,  and  the  pitch  has  passed  into  the  Octave  above.  Tlie  quality 
of  the  lowest  tone  of  the  string  is  rendered  softer  and  brighter,  but  less  fidl  and 
powerful  when  the  intermixture  commences.  It  is  interesting  to  observe  the 
vi1)rational  figure  while  little  changes  are  made  in  the  style  of  bowing,  and  note 
how  tlie  resulting  slight  changes  of  quality  are  immedi.itely  rendered  evident  by  H 
verv  distinct  changes  in  the  vibrational  figure  itself. 

The  vibrational  forms  just  described  may  be  maintained  in  a  unifonnly  steady 
and  unchanged  condition  by  carefully  uniform  bowing.  The  instrument  has  then 
an  uninterrupted  and  pure  musical  quality  of  tone.  Any  scratching  of  the  bow  is 
immediately  shown  by  sudden  jumps,  or  discontinuous  displacements  and  changes 
in  the  vibrational  figure.  If  the  scratching  continues,  the  eye  has  no  longer  time 
to  perceive  a  regular  figure.  The  scratching  noises  of  a  violin  bow  must  therefore 
be  regarded  as  irregular  interruptions  of  the  normal  vibrations  of  the  string, 
making  them  to  recommence  from  a  new  starting  jioint.     Sudden  jumps  in  the 


vibrational  figure  betray  every  little  stumble  of  the  bow  which  tlie  ear  alone  would 
scarcely  observe.  Inferior  bowed  instrvmients  seem  to  be  distinguished  from  good 
ones  by  the  freciuency  of  such  greater  or  smaller  irregularities  of  vibration.  On 
the  string  of  my  monochord,  which  was  only  used  for  the  occasion  as  a  bowed 
instrument,  great  neatness  of  bowing  was  required  to  preserve  a  steady  vibrational 
figure  lasting  long  enough  for  the  eye  to  apprehend  it ;  and  the  tone  was  rough  in 
(piality,  accompanied  by  much  scratching.  With  a  very  good  modern  violin  made 
by  Bausch  it  was  easier  to  maintain  the  steadiness  of  the  vibrational  figure  for 
some  time ;  but  I  succeeded  much  better  with  an  old  Italian  violin  of  Guadanini, 
which  was  the  first  one  on  which  I  could  keep  the  vibrational  figure  steady  enough  H 
to  count  the  crumples.  This  great  uniformity  of  vibration  is  evidently  the  reason 
of  the  purer  tone  of  these  old  instruments,  since  every  little  irregularity  is  imme- 
diately felt  by  the  ear  as  a  roughness  or  scratchiness  in  the  quality  of  tone. 

An  appropriate  structure  of  the  instrument,  and  wood  of  the  most  perfect 
elasticity  procurable,  are  probably  the  important  conditions  for  regular  vibrations 
of  the  string,  and  when  these  are  present,  the  bow  can  be  easily  made  to  work 
uniformly.  This  allows  of  a  pure  flow  of  tone,  undisfigured  by  any  roughness. 
On  the  other  hand,  when  the  vibrations  are  so  tuiiform  the  string  can  be  more 
vigorously  attacked  with  the  bow.  Good  instruments  consequently  allow  of  a  much 
more  powerful  motion  of  the  string,  and  the  whole  intensity  of  their  tone  can  be 
communicated  to  the  air  without  diminution,  whereas  the  friction  caused  by  any 
imperfection  in  the  elasticity  of  the  wood  destroys  part  of  the  motion.  Much  of 
the  advantages  of  old  violins  may,  however,  also  depend  upon  their  age,  and  espe- 
cially their  long  use,  both  of  which  cannot  but  act  favoiu-ably  on  the  elasticity  of 


86  MUSICAL  TONES  OF  BOWED  INSTRUMENTS.  part  i. 

the  wood.  But  the  art  of  bowing  is  evidently  the  most  important  condition  of  all. 
How  delicately  this  must  he  cultivated  to  obtain  certainty  in  producing  a  very 
perfect  quality  of  tone  and  its  difFei-ent  varieties,  cannot  be  more  cleaidy  demon- 
strated than  by  the  observation  of  vibrational  figiires.  It  is  also  well  known  that 
great  players  can  bring  out  full  tones  from  even  indifferent  instruments. 

The  preceding  observations  and  conclusions  refer  to  the  vibrations  of  the  strings 
of  the  instrument  and  the  intensity  of  their  upper  partial  tones,  solely  in  so  far  as 
they  are  contained  in  the  compound  vibrational  movement  of  the  string.  But 
partial  tones  of  different  pitches  are  not  equally  well  communicated  to  the  air,  and 
hence  do  not  strike  the  ear  of  the  listener  with  precisely  the  same  degrees  of 
intensity  as  those  they  possess  on  the  string  itself.  They  are  communicated  to 
the  air  by  means  of  the  sonorous  body  of  the  instrument.  As  we  have  had 
already  occasion  to  remark,   vibrating  strings  do  not   directly   communicate   any 

^sensible  portion  of  their  motion  to  the  air.  The  vibrating  strings  of  the  violin, 
in  the  first  place,  agitate  the  bridge  over  which  they  are  stretched.  This  stands 
on  two  feet  over  the  most  mobile  part  of  the  'belly'  between  the  two  '/  holes'. 
One  foot  of  the  bridge  rests  upon  a  comparatively  firm  support,  namely  the  '  sound- 
post,'  which  is  a  solid  rod  inserted  between  the  two  plates,  back  and  belly,  of  the 
instrument.  It  is  only  the  other  leg  which  agitates  the  elastic  wooden  plates,  and 
through  them  the  included  mass  of  air.* 

An  inclosed  mass  of  air,  like  that  of  the  violin,  viola,  and  violoncello,  bounded 
by  elastic  plates,  has  certain  proper  tones  which  may  be  evoked  by  l)lowing 
across  the  openings,  or  '/  holes '.  The  violin  thus  treated  gives  c'  according  to 
Savart,  wdio  examined  instruments  made  by  Stradivari  (Stradiuarius).t  Zam- 
miner  found  the  same  tone  constant  on  even  imperfect  instruments.  For  the 
violoncello  Savart  found  on  blowing  over  the  holes  F,  and  Zamminer  0.%  Ac- 
cording to  Zamminer  the  soimd-box  of  the  viola   (tenor)   is  tuned  to  be  a  Tone 

•H deeper  than  that  of  the  violin. §  On  placing  the  ear  against  the  back  of  a  violin 
and  playing  a  scale  on  the  pianoforte,  some  tones  will  be  found  to  penetrate  the 
ear  with  more  force  than  others,  owing  to  the  resonance  of  the  instrument.     On  a 


*  [This   account   is    not    quite    sufficient.  agitation  ti-ansmitted  by  the  rod."      In  short, 

Neither   leg   of   the   bridge   rests   exactly  on  the  touch  rod  acts  as   a   sound-post   to   the 

the  sound-post,  because  it  is  found  that  this  finger.      The  place  of  least  vibration  of  the 

position  materially  injures  the  quality  of  tone.  belly  is  exactly  over  the  sound-post  and  of  the 

The  sound-post  is  a  little  in  the  rear  of  the  back  at  the  point  under  the  sound-post.     On 

\eo  of  the  bridge  on  the  v"  string  side.     The  removing  the  sound-post,  or  covermg  its  ends 

pcTsition  of  the  sound-post  with  regard  to  the  with  a  sheet  of  india-rul)ber,  which  did  not 

bridge  has  to  be  adjusted  for  each  individual  diminish  the  support,  the  tone  was  poor  and 

instrument.     Dr.  William  Muggins,  F.R.S.,  in  thin.     But  an  external  wooden  clamp  connect- 

his  paper  '  On  the  Function  of  the  Sound-post,  ing  belly  and  back  in  the  places  where  the 

and   on   the   Proportional   Thickness   of    the  sound-post  touches  them,  restored  the  tone.— 

Strings   of  the  Violin,'  read   Mav   24,    188.3,  Translator.] 

ProceecUnqs   of  thr    Koi/al    Society,   vol.    xxxv.  t  [Zamminer,    Die    Musik,    1855,    vol.    i. 

t|  pp.  241-248,  "has  experimentally  investigated  p.  37,  says  c'  of  256  \ih.— Translator.'] 

the  whole  action  of  the  sound-post,  and  finds  +  [Zamminer,  ihid.  p.    41,  and  adds  that 

that  its  main  function  is  to  convey  vibrations  judging  from  the  violm  the  resonance  should 

from  the  belly  to  the  back  of  the  violin,  in  he  F%.— Translator.] 

addition  to  those  conveyed  by  the  sides.     The  §    [The  passage  referred  to  has  not  been 

(apparently  ornamental)  cuttings  in  the  bridge  found.      But    Zammmer    says,   p.    40,    'The 

of  the  viohn,  sift  the  two  sets  of  vibrations,  length  of  the  box  of  a  violin  is  13  Par.  inches, 

set  up  by  the  bowed  string  at  right  angles  to  and  of  the  viola  14  inches  5  lines.     Exactly 

each  other  and  '  allow  those  only  or  mainly  to  in   inverse   ratio   stand    the    pitch    numbers 

pass  to  the  feet  which  would  be  efficient  in  470  (a  misprint  for  270  most  probably)  and 

setting  the  body  of  the  instrument  into  vibra-  241,  which  were  found  by  blowing  over  the 

tion'.      As  the  peculiar  shape  of  the  instru-  wind-holes  of  the  two  instruments.'     Now  the 

ment  rendered  strewing  of  sand  unavailable,  ratio  13 :  14t%  gives  182  cents,  and  the  ratio 

Dr.  Huggins   investigated   the  vibrations  by  241 :    270  gives   197   cents,   which    are    very 

means  of  a  '  touch  rod,'  consisting  of  '  a  small  nearly,  though  not  '  exactly  '  the  same.     This, 

round   stick   of   straight   grained   deal  a  few  however,  makes  the  resonance  of  the  violm 

inches  long  ;  the  forefinger  is  placed  on  one  270  vib.  and  not  256  vib.,  and  agrees  with  the 

end  and  the  other  end  is  put  Ughtly  in  contact  next  note.     I  got  a  good  resonance  with  a  fork 

with  the  vibrating  surface.     The  'finger  soon  of  268  vib.  from  Dr.  Muggins's  violoncello  by 

becomes  very  sensitive  to  small  differences  of  Nicholas  about  a.d.  Vl<d±^Translator.] 


MUSICAL  TONES  OF    HOWKD   INSTRUMENTS. 


87 


violin  made  by  Bauscli  two  tones  of  greatest  resonance  were  thus  discovered,  one 
l)etwcen  r'  and  c'Z  [between  264  and  280  vib.],  and  the  other  between  a  and  6'|j 
[between  440  and  466  vib.].  For  a  vi(>la  (tenor)  1  found  tlie  two  tones  about  a 
Tone  deeper,  which  agrees  with  Zaniminer's  calculation.* 

The  consequence  of  this  peculiar  relation  of  resonance  is  that  those  tones  of 
the  strings  which  lie  near  the  proper  tones  of  the  inclosed  body  of  air,  must  be 
proportionably  more  reinforced.  This  is  clearly  perceived  on  both  the  violin  and 
violoncello,  at  least  for  the  lowest  proper  tone,  when  the  corresponding  notes  are 
produced  on  the  strings.  They  sound  particularly  full,  and  tlie  jn-ime  tone  of  these 
compoiinds  is  especially  prominent.  I  think  that  I  heai'd  this  also  for  a'  on  the 
violin,  which  corresponds  to  its  higher  proper  tone. 

Since  the  lowest  tone  on  the  violin  is  g,  the  only  upper  partials  of  its  musical 
tones  which  can  be  somewhat  reinforced  l)y  the  resonance  of  the  higher  proper 
tone  of  its  inclosed  body  of  air,  are  the  higher  octaves  of  its  three  deepest  notes.  II 
Hnt  the  prime  tones  of  its  higher  notes  will  be  reinforced  more  than  their  upper 
partials,  because  these  prime  tones  are  more  nearly  of  the  same  pitch  as  the 
proper  tones  of  the  body  of  air.  This  produces  an  effect  similar  to  that  of  the  con- 
struction of  the  hammer  of  a  piano,  which  favours  the  upper  partials  of  the  deep 
notes,  and  weakens  those  of  the  higher  notes.  For  the  violoncello,  where  the  lowest 
string  gives  C,  the  stronger  proper  tone  of  the  body  of  air  is,  as  on  the  violin,  a 
Fourth  or  a  Fifth  higher  than  tlie  pitch  of  the  lowest  string.  There  is  consequently 
a  similar  ivlation  between  the  favoured   and    unfavoured   partial  tones,  but  all  of 


*  rThrough  the  kiuduess  of  Dr.  Huggius, 
F.R.S.,  the  Rev.  H.  R.  Haweis,  and  the  violin- 
makers.  Messrs.  Hart,  Hill  &  Withers,  I  was 
in  18S0  enabled  to  examine  the  pitch  of  the 
resonance  of  some  fine  old  violins  by  Duiff'o- 
prugcar  (Swiss  Tyrol,  Bologna,  and  Lyons 
1510-1538),  Amati  (Cremona  1596-1684),  Rug- 
gieri  (Cremona  1668-1720),  Stradivari  (Cre- 
mona 1644-1737),  Giuseppe  Garneri  (known  as 
'  Joseph,'  Cremona  1683-1745),  Lupot  (France 
1750-1820).  The  method  adopted  was  to  hold 
tuning-forks,  of  which  the  exact  pitch  had 
been  determined  by  Scheibler's  forks,  in  succes- 
sion over  the  widest  part  of  the  /  hole  on  the 
(J  string  side  of  the  violin  (furthest  from  the 
sound-iDost)  and  observe  what  fork  excited  the 
maximum  resonance.  jMy  forks  form  a  series 
proceeding  by  4  vib.  in  a  second,  and  hence  I 
could  only  tell  the  pitch  within  2  vib.,  and  it 
was  often  extrenrely  difficult  to  decide  on  the 
fork  which  gave  the  best  resonance.  By  far 
the  strongest  resonance  lay  between  208  and 
272  vib.,  but  one  early  Stradivari  (1696)  had  a 
fine  resonance  at  264  vib.  There  was  also  a 
secondary  but  weaker  maximum  resonance  at 
about  252  vib.  The  256  vib.  was  generally 
decidedly  inferior.  Hence  we  may  take  270 
vib.  as  tile  primary  maximum,  and  252  vib.  as 
the  secondary.  The  first  corresponds  to  the 
liighest  English  concert  pitch  .■"  =  540  vib., 
now  used  in  London,  and  agrees  with  the 
lower  resonance  of  Bausch's  instrument  men- 
tioned in  the  text.  The  second,  which  is  120 
cents,  or  rather  more  than  an  equal  Semitone 
flatter,  gives  the  pitch  which  my  researches 
show  was  common  over  all  Europe  at  the 
time  (see  App.  XX.  sect.  H.).  But  although 
the  low  pitch  was  prevalent,  a  high  pitch,  a 
great  Semitone  (117  ct.)  higher,  was  also  in 
use  as  a  '  chamber  pitch '.  A  violin  of  Mazzini 
of  Brescia  (1560-1640)  belonging  to  the  eldest 
daughter  of  ilr.  Vernon  Lushiugton,  Q.C.,  had 
the  same  two  maximum  resonances,  the  higher 
being  decidedly  the  superior.     I  did  not  ex- 


amine for  the  higher  or  «'  pitches  named  in 
the  text.  IMr.  Healey  (of  the  Science  and  Art 
Department,  South  Kensington)  thought  this 
violin  (supposed  to  be  an  Amati)  sounded  best 
at  the  low  pitch  c"  =  504.  Subsequently,  I  ex- 
amined a  fine  instrument, 'bearing  inside  it  the 
label  '  Petrus  Guarnerius  Cremonensis  fecit, 
Mantuse  sub  titulo  S.  Theresiae,  anno  1701,'  in 
the  possession  of  Mr.  A.  J.  Hipkins,  who  knew  ^ 
it  to  be  genuine.  I  tried  this  with  a  series 
of  forks,  proceeding  by  differences  of  about 
4  vibrations  from  240  to  560.  It  was  surprising 
to  find  that  every  fork  was  to  a  certain  extent 
reinforced,  that  is,  in  no  case  was  the  tone 
quenched,  and  in  no  case  was  it  reduced  in 
strength.  But  at  260  vib.  there  was  a  good, 
and  at  264  a  better  resonance  ;  perhaps  262 
may  therefore  be  taken  as  the  best.  There 
was  no  secondary  low  resonance,  but  there 
were  two  higher  resonances,  one  about  472, 
(although  468  and  476  were  also  good),  and 
another  at  520  (although  524  and  528  were 
also  good).  As  this  sheet  was  passing  through 
the  press  I  had  an  opportunity  of  trying  the 
resonance  of  Dr.  Huggins's  Stradivari  of  1708, 
figured  in  Grove's  JJictionari/  of  Music,  iii. 
728,  as  a  specimen  of  the  best  period  of  Stradi- 
vari's work.  The  result  was  essentially  thefl 
same  as  the  last ;  every  fork  was  more  or  less 
reinforced  ;  there  was  a  subordinate  maximum 
at  252  vib.  ;  a  better  at  froiu  260  to  268  vib.  ; 
very  slight  maxima  at  312,  348,  384,  412,  420, 
428  (the  last  of  which  was  the  best,  but  was 
only  a  fair  reinforcement),  472,  to  480,  but  520 
was  decidedly  best,  and  540  good.  No  one 
fork  was  reinforced  to  the  extent  it  would  have 
been  on  a  resonator  properly  tuned  to  it,  but 
no  one  note  was  deteriorated.  Dr.  Huggins  says 
that  '  the  strong  feature  of  this  violin  is  the 
great  equality  of  all  four  strings  and  the  per- 
sistence of  the  same  fine  quality  of  tone 
throughout  the  entire  range  of  the  instru- 
ment '. — Trail  a]  (dor.  1 


88  MUSICAL  TONES  OF  FLUTE  OR  FLUE  PIPES.  part  i. 

them  are  a  Twelfth  lower  than  on  the  violin.  On  the  other  hand,  the  most 
favoured  partial  tones  of  the  vi(jla  (tenor)  corresponding  nearly  with  //,  do  not 
lie  between  the   first  and    second  strings,   but  ^^^^  _^^ 

between  the  second  and  third;  and  this  seems  •'  ^ 

to  be  connected  with  the  altered  quality  of 
tone  on  this  instrument.  Unfortunately  this 
influence  cannot  be  expressed  numerically. 
The  maximum  of  resonance  for  the  proper 
tones  of  the  body  of  air  is  not  very  marked  ; 
were  it  otherwise  there  would  be  much  more 
inequality  in  the  scale  as  played  on  these 
bowed  instruments,  immediately  on  passing 
the  pitch  of  the  proper  tones  of  their  bodies  of 
Hair.  We  must  consequently  conjecture  that 
their  influence  upon  the  relative  intensity  of  | 
the  individual  partials  in  the  musical  tones  of  [ 
these  instruments  is  not  very  prominent.  j 

5.   Musical  Tones  of  Flute  or  Flue  Pipes. 
In  these  instruments  the  tone   is  produced 

by  driving  a  stream  of  air  against  an  opening, 

generally  furnished  with  sharp  edges,  in   some 

hollow    space   filled    with    air.       To    this    class 

belong  the  bottles  described  in  the  last  chapter, 

and   shown  in  fig.    20  (p.   60c),   and   especially 

flutes    and  the   majority   of   organ    pipes.     For 

flutes,  the  resonant  body  of  air  is  included  in 
H  its  own  cylindrical  bore.     It  is  blown  with  the 

mouth,    which   directs   the    breath   against  the 

somewhat   sharpened  edges  of   its  mouth  hole. 

The  construction  of   organ   pipes   will  be    seen 

from  the    two   adjacent   figures.      Fig.   27,    A, 

shows   a   square   w^ooden   pipe,   cut  open   long- 
wise, and  B  the  external  appearance  of  a  round 

tin  pipe.     R  E.  in  each  shows  the  tube  which 

incloses  the  sonorous  body   of   air,    a    b  is  the 

rtumth  where  it  is  blown,  terminating  in  a  sharp 

lip.     In  fig.   27,  A,   we  see  the  air  chamber  or 

throat  K  into  Avhich  the  air  is  first  driven  from 

the   bellows,    and   whence    it    can    only    escape 

through    the    narrow  slit  c  d,   which  directs  it 
11  against  the  edge  of  the  lip.     The  w^ooden  pipe 

A  as  here  drawn  is  open,  that  is  its  extremity 

is   uncovered,    and   it  produces  a  tone   with   a 

wave   of  air   tivic.e   as  long  as  the  tube    R    R. 

The  other  pipe,  B,  is  stopped,  that  is,  its  upper 

extremity  is  closed.     Its  tone  has  a  wave  four 

times  the  length  of  the  tube  R  R,  and  hence  an 

Octave  deeper  than  an   open  pipe  of  the  same 

length.* 

Any  air  chambers  can    be    made    to  give  a 

musical  tone,  just  like   organ   pipes,    flutes,  the   bottles   previously  described,   the 

windchests    of    violins,    Ac,    provided    they    have    a    sufficiently    narrow    opening, 


*  [These  relations  are  only  approximate, 
as  is  explained  below.  The  mode  of  excite- 
ment  by   the   lip   of   the   pipe   makes    them 


inexact.  Also  they  take  no  notice  of  the 
'  scale  '  or  diameters  and  depths  of  the  pipes, 
or  of  the  force  of  the  wind,  or  of  the  tempera- 


CHAP.  V.  5.         MUSICAL  TONES  OF  FLUTK  Oil   FLIK   I'lPKS.  S9 

furnished  with  somewhat  projecting  sharp  eilges,  l)y  ilirectiiig  a  tliiii   Hat  stream   of 
air  across  the  opening  and  against  its  edges.* 

The  motion  of  air  that  takes  place  in  the  inside  of  organ  i)ipes,  corresponds  to 
a  system  of  plane  waves  which  are  reflected  backwards  and  forwards  between  the 
two  ends  of  the  pipe.  At  the  stopj^ed  end  of  a  cylindrical  pipe  the  reflexion  of 
every  wave  that  strikes  it  is  very  perfect,  so  that  the  reflected  wave  has  the  same 
intensity  as  it  had  before  reflexion.  In  any  train  of  waves  moving  in  a  given 
direction,  the  velocity  of  the  oscillating  molecules  in  the  condensed  portion  of  the 
wave  takes  place  in  the  same  direction  as  that  of  the  propagation  of  the  waves,  and 
in  the  rareHed  portion  in  the  opposite  direction.  But  at  the  stopped  end  of  a  pipe 
its  cover  does  not  allow  of  any  forward  motion  of  the  n\oleculcs  of  air  in  the 
direction  of  the  length  of  the  |)ipe  Hence  the  incident  and  reflected  wave  at  this 
place  combine  so  as  to  excite  opposite  velocities  of  oscillation  of  the  molecxiles  of 
air,  and  consequently  by  their  superposition  the  velocity  of  the  molecules  of  air  at  H 
the  cover  is  destroyed.  Hence  it  follows  that  the  phases  of  pressure  in  both  will 
agree,  because  opposite  motions  of  oscillation  and  opposite  propagation,  result  in 
accordant  pressure. 

Hence  at  the  stopped  end  tliere  is  no  motion,  but  great  alteration  of  pressure. 
The  reflexion  of  the  wave  takes  place  in  such  a  manner  that  the  phase  of  conden 
sation  remains  unaltered,  but  the  direction  of  the  motion  of  oscillation  is  reversed. 

The  contrary  takes  place  at  the  open  end  of  pipes,  in  which  is  also  included  the 
<ipening  of  their  mouths.  At  an  open  end  where  the  air  of  the  pipe  connnuni- 
cates  freely  with  the  great  outer  mass  of  air,  no  sensible  condensation  can  take 
place.  In  the  explanation  usually  given  .of  the  motion  of  air  in  pipes,  it  is  assumed 
that  both  condensation  and  rarefaction  vanish  at  the  open  ends  of  pipes,  which  is 
aijproximately  but  not  exactly  correct.  If  there  were  exactly  no  alteratit)n  of 
density  at  that  place,  there  would  be  complete  reflexion  of  every  incident  wave 
at  the  open  ends,  so  that  an  equally  large  reflected  wave  would  be  generated  with  H 
an  opposite  state  of  density,  but  the  direction  of  oscillation  of  the  molecules  of 
air  in  both  waves  would  agree.     The  superposition  of  such  an  incident  and  such  a 


ture  of  the  air.      The  following  are  adapted  from    2f    to    3.^    inches,    the   pitch    number 

from  the  rules  given  by  51.  Cavaille-Coll,  the  increases  bj'  about  1  in  300,  but  as  pressure 

celebrated  French  organ-builder,  in   Comjites  varies  from  3^  to  4  inches,  the  pitch  number 

Rciidus,  1860,  p.  176,  supposing  the  tempera-  increases  by  about  1  in  440,  the  whole  increase 

ture  to  be  59'  F.  or  15"  C,  and  the  pressure  of  of  pressure  from  23  to  4  inches  increases  the 

the  wind  to  be  about  3^  inches,  or  8  centi-  pitch  number  by  1  in  180. 
metres  (meaning  that  it  will  support  a  column  For   temperature,    I   found   by   numerous 

of  water  of  that  height  in  the  wind  gauge).  observations   at    very   different    temperatures 

The  pitch  niunbers,  for  donhle  vibrations,  are  that  the  following  practical  rale  is  sufficient 

found  by  dividing  20,080  when  the  dimensions  for  reducing  tbe  pitch  number  observed  at  one 

ai-e   given   in   inches,    and    510,000   when   in  temperature  to  that  due  to  another.     It  is  not 

millimetres  by  the  following  numbers :  (1)  for  quite  accurate,   for  the  air  Ijlown    from   the 

ei/lindrical  open    pipes,    3    times    the    length  bellows  is  often  lower  than  the  external  tem- 

added  to  5  times  the  diameter ;  (2)  for  cfjliiidyi-  perature.     Let  F  be  the  pitch  number  observed  ^ 

ad  stopped  pipes,  G  times  the  length  added  to  at  a  given  temperature,  and  d  the  difference  of 

10  times  the  diameter;    (3)    for   square  open  temperature  in  degrees  Fahr.     Then  the  pitch 

pipes,  3  times  the  length  added  to  6  times  the  number  is  P  x  (1  +    00104  d)  according  as  the 

depth  (clear  internal  distance  from  mouth  to  temperature  is  higher  or  lower.     Tbe  practical 

back  ;  (4)  for  squnre  stopped  pipes,  6  times  the  operation  is  as  follows :  supposing  P  =  528,  and 

length  added  to  12  times  the  depth.  d  =  14  increase  of  temperature.     To  528  add 

This  rule  is  always  sufficiently  accurate  for  4  in  100,  or  21-]2,  giving  549-12.     Divide  by 

cutting    organ    pipes    to    their    approximate  1000    to    2    places    of   decimals,    giving   -55. 

length,  and  piercing  them  to  bring  out  the  :Multiply  by  J  =  14,  giving  7-70.     Adding  this  to 

(Jctave  harmonic,  and  has  long  been  used  for  528,  we  get  535-7  for  the  pitch  number  at  the 

these  purposes  in  M.  Cavaille-CoU's    factory.  new  temperature.— 7'y-r^^«^f/o/•.] 
The  rule  is  not  so  safe  for  the  square  wooden  *  [Here  the  passage  from  '  These   edges,' 

as  for  the  cylindrical  metal  pipes.     The  pitch  p.  140,  to  'resembling  a  violin,'  p.  141  of  the 

of  a  pipe  of  known  dimensions  ought  to  be  1st  Enghsh  edition,  has  been  omitted,  and  the 

tirst  ascertained  by  other  means.     Then  this  passage    from    'The   motion   of   air,'   p.   89rt, 

pitch  number  multiplied  by  the  divisors  in  (3)  to  '  their  corners  are  rounded  oi?,'  p.  93?^,  has 

and  (4)  should  be  used  in  place  of  the  20,080  been   inserted   in   accordance   with    the    4th 

or  510,000  of  the  rule,  for  all  similar  pipes.  German  edition.—  Tran slat m:^ 

As  to  strength  of  wind,  as  pressure  varies 


90  MUSICAL  TONES  OF  FLUTE  OR  FLUE  PIPES.  pakt  i. 

reflected  wave  would  indeed  leave  the  state  of  density  unaltered  at  the  open  ends, 
hut  would  occasion  great  velocity  in  the  oscillating  molecules  of  air. 

In  reality  the  assumption  made  explains  the  essential  phenomena  of  organ  pipes. 
Consider  first  a  pipe  with  two  open  ends.  On  our  exciting  a  wave  of  condensation, 
at  one  end,  it  runs  forward  to  the  other  end,  is  there  reflected  as  a  wave  of  rare- 
faction, runs  back  to  the  first  end,  is  here  again  reflected  with  another  alteration  of 
phase,  as  a  wave  of  condensation,  and  then  repeats  the  same  path  in  the  same  way 
a  second  time.  This  repetition  of  the  same  process  therefore  occurs,  after  the 
wave  in  the  tube  has  passed  once  forwards  and  once  l)ackwards,  that  is  twice  through 
the  whole  length  of  the  tube.  The  time  required  to  do  this  is  equal  to  double  the- 
length  of  the  pipe  divided  by  the  velocity  of  somid.  This  is  the  diiration  of  the 
vibration  of  the  deepest  tone  which  the  pipe  can  give. 

Suppose  now  that  at  the  time  when  the  wave  begins  its  second  forward  and 
%  backward  journey,  a  second  impulse  in  the  same  direction  is  given,  say  by  a  vibra- 
ting tuning-fork.     The  motion  of  the  air  will  then  receive  a  reinforcement,  which 
will  constantly  increase,  if  the  fresh  impulses  take  place  in  the  same  rhythm  as  the 
forward  and  backward  progression  of  the  waves. 

Even  if  the  returning  wave  does  not  coincide  with  the  first  following  similar 
impulse  of  the  tuning-fork,  but  only  with  the  second  or  third  or  foiu-th  and  so  on,, 
the  motion  of  the  air  will  be  reinforced  after  every  forward  and  l)ackward  passage. 

A  tube  open  at  both  ends  will  therefore  serve  as  a  resonator  for  tones  whose 
pitch  number  is  equal  to  the  velocity  of  sound  (332  metres)  *  divided  by  twice  the 
length  of  the  tube,  or  some  multiple  of  that  number.  That  is  to  say,  the  tones  of 
strongest  resonance  for  such  a  tube  will,  as  in  strings,  form  the  complete  series  of 
harmonic  upper  partials  of  its  prime. 

The  case  is  somewhat  difterent  for  pipes  stopped  at  one  end.  If  at  the  open 
end,  l.'y  means  of  a  vibrating  tuning-fork,  we  excite  an  impulse  of  condensation 
^  which  propagates  itself  along  the  tube,  it  Avill  run  on  to  the  stopped  end,  will  be 
there  reflected  as  a  wave  of  condensation,  return,  will  be  again  reflected  at  the 
open  end  with  altered  phase  as  a  Avave  of  rarefaction,  and  only  after  it  has  been 
again  reflected  at  the  stopped  end  with  a  similar  phase,  and  then  once  more  at  the 
open  end  with  an  altered  phase  as  a  condensation,  will  a  repetition  of  the  process 
ensue,  that  is  to  say,  not  till  after  it  has  traversed  the  length  of  the  pipe  four  times. 
Hence  the  prime  tone  of  a  stopped  pipe  has  twice  as  long  a  period  of  vibration  as  an 
open  pipe  of  the  same  length.  That  is  to  say,  the  stopped  pipe  will  be  an  Octave 
deeper  than  the  open  pipe.  If,  then,  after  this  double  forward  and  backward  passage, 
the  first  impiilse  is  renewed,  there  will  arise  a  reinforcement  of  resonance. 

Partials  f  of  the  prime  tone  can  also  be  reinforced,  but  only  those  which  are 
unevenly  numbered.  For  since  at  the  expiration  of  half  the  period  of  vibration, 
the  prime  tone  of  the  wave  in  the  tube  renews  its  path  with  an  opposite  phase  of 
density,  only  such  tones  can  be  reinforced  as  have  an  opposite  phase  at  the  expira- 
^tion  of  half  the  period  of  vibration.  But  at  this  time  the  second  partial  has  just 
completed  a  whole  vibration,  the  fourth  partial   two  whole  vibrations,  and  so  on. 

*  [This  is  3089-3  feet  in  a  second,  which  l)efore  the    Physical    Society,   and   pubUshed 

is  the  mean  of  several  observations  in  free  in  the  Philoso})hical  Maqazine  for  Dec.  1883, 

air;  it  is  usual,  however,  in  England  to  take  pp.  447-455,  and  Oct.  1884,  pp.  328-834,  as  the 

the  whole  number  1090  feet,  at  freezing.     .\t  means  of  many  observations  on  the  velocity 

60"'  F.  it  is  about  1120  feet  per  second.     Mr.  D.  of  sound  in  dry  air  at  32°  F.,  in  tubes,  obtained 
J.  Blaikley  (see  note  p.  97(0!  ii''  two  papers  read 

for  diameter  -45  -75  1-25  2-08  347  English  inches, 

pitch  various,  velocity        1064-26        1072-53        1078-71        1081-78       1083-13        „        feet. 
pitch  260  vib.,  velocity      1062-12        1072-47        1078-73        1082-51        1084-88 

Tlie  velocity  in  tubes  is  therefore  always  less  note  p.  23t-),  but  it  is  precisely  the  latter  which 

than  in  free  air.—  TrcDnt/afor.]  are  not  excited  in  the  present  case.     This  is 

t  [The     original     says     '  upper     partials '  only  mentioned  as  a  warning  to  those  who 

(Obertoii'^),  but  the  vpper  partials  which  are  faultily  use  the  faulty  expression  '  overtones  ' 

unevenly  numbered  are  the  1st,  3rd,  5th,  &c.,  indifferently    for    both    partials    and    vjrper 

and  these  are  really  the  2nd,  4th,  6th,  &c.,  (that  partials.^ 7'v-«//.s-A(('f)r.] 
is,  the  evenly  numbered)  partial  tones  (see  foot- 


CHAP.  V.  5.         MUSICAL  TONES  OF  FLUTE  Oil  FLUE  I'LPES.  91 

These  therefore  have  the  same  phases,  and  cancel  their  etiect  ou  the  return  of  the 
wave  with  an  opposite  phase.  Hence  the  tones  of  strongest  resonance  in  sto[)ped 
jjipes  correspond  with  the  series  of  unevenly  numbered  partials  of  its  fundamental 
tone.  Supposing  its  pitch  number  is  n,  then  3/?,  is  the  Twelfth  of  u,  that  is  the 
Fifth  of  -In  the  higher  Octave,  and  5/;  is  the  major  Third  of  \n  tlie  next  liigher 
Octave,  and  In  the  [sub]  minor  Seventh  of  the  same  Octave,  and  so  on. 

Now  although  the  phenomena  follow  these  rules  in  the  principal  points,  certain 
deviations  from  them  occur  because  there  is  not  precisely  no  change  of  pressiu-e 
at  the  open  ends  of  pipes.  From  these  ends  the  motion  of  sound  communicates 
itself  to  the  uninclosed  air  beyond,  and  the  waves  which  spread  out  from  the  open 
ends  of  the  tubes  have  relatively  very  little  alteration  of  pressure,  but  are  not 
entirely  without  some.  Hence  a  part  of  every  wave  which  is  incident  on  the  open 
end  of  tlie  pipe  is  not  reflected,  but  runs  out  into  the  open  air,  while  the  remainder 
or  greater  portion  of  the  wave  is  reflected,  and  returns  into  the  tube.  The  re-H 
flexion  is  the  more  complete,  the  smaller  are  the  dimensions  of  the  opening  of 
the  tube  in  comparison  with  the  wave-length  of  the  tone  in  question. 

Theory*  also,  agreeing  with  experiment,  shows  that  the  phases  of  the  reflected 
part  of  the  wave  are  the  same  as  they  would  be  if  the  reflexion  did  not  take  place 
at  the  surface  of  the  opening  itself  but  at  another  and  somewhat  ditterent  plane. 
Hence  what  may  be  called  the  reduced  length  of  the  pipe,  or  that  answering  to  the 
pitch,  is  somewhat  different  from  the  real  length,  and  the  difference  between  the 
two  depends  on  the  form  of  the  mouth,  and  not  on  the  pitch  of  the  notes  pro- 
duced unless  they  are  so  high  and  hence  their  wave-lengths  so  short,  that  the 
dimensions  of  the  opening  cannot  be  neglected  in  respect  to  them. 

For  cylindrical  pipes  of  circular  section,  with  ends   cut  at  right  angles  to  the 
length,  the  distance  of  the  plane  of  reflexion  from  the  end  of  the  pipe  is  theoreti- 
cally determined  to  be  at  a  distance  of  0-785-t   the   radius  of  the  circle.!     For  a 
wooden  pipe  of  square  section,  of  which  the  sides  were  36  mm.  (1-4  inch)  internal  II 
measure,  I  found  the  distance  of  the  plane  of  reflexion  14  mm.  (-55  inch).^ 

Now  since  on  account  of  the  imperfect  reflexion  of  waves  at  the  open  ends  of 
organ  pipes  (and  respectively  at  their  mouths)  a  part  of  the  motion  of  the  air  must 
escape  into  the  free  air  at  every  vibration,  any  oscillatory  motion  of  its  mass  of  air 
must  be  speedily  exhausted,  if  there  are  no  forces  to  replace  the  lost  motion.  Li 
fact,  on  ceasing  to  blow  an  organ  pipe  scarcely  any  after  sound  is  observable. 
Nevertheless  the  wave  is  frequently  enough  reflected  forward  and  backward  for  its 
pitch  to  become  perceptible  on  tapping  against  the  pipe. 

The  means  usually  adopted  for  keeping  them  continually  sounding,  is  hloirlag. 
In  order  to  understand  the  action   of  this  jirocess,  we  must  remember  tliat   when 

"  See  my    paper    ui    Crellcn    Jdunud  for  tion  of  the  plug.     [The  sameness  of  the  pitch 

Mathematics,  vol.  Ivii.  is  determined  by  seeiug  that  each  makes  the 

t  Mr.  Bosanquc't  {Proc.  Mas.  Assn.  1877-8,  same  number  of  beats  with  the  same  fork.] 

p.65)  is  reported  as  saying: 'Lord  Rayleigh  and  The  nodal   surface  lay  137  mm.    (5-39  inch) 

himself  had  gone  fully  into  the  matter,  and  had  from  the  end  of  the  pipe,  while  a  quarter  of  II 

come  to  the  conclusion  that  this  correction  was  a  wave-length  was  151  mm.  (5-94  inch).   At  the 

much  less  than  Helmlioltz  supposed.  Lord  Ray-  mouth  end  of  the  pipe,  on   the  other   hand, 

leigh  adopted  tlie  figure  -6  of  the  radius,  whilst  83  mm.  (3-27  inch)  were  wanting  to  complete 

he  himself  adopted  -55. '     See  papers  by  Lord  the  theoretical  length  of  the  pipe.      [The  addi- 

Rayleigh  and  Mr.  Bosanquet  in  Philosophical  tional  piece  l)eing  half  the  length  of  the  wave, 

Mayazine.      i\Ir.   Blaikley   by   a   new   process  the   pitch   of   the  pipe  before  and   after  the 

finds  -576,  which  lies  between  the  other  two,  addition  of  this  piece  remains  the  same,  by 

see  his  paper  in  Phil.  Mag.  Mav  1879,  p.  342.  which  propeity  the  length  of  the  additional 

+  The  pipe  was  of  wood,  made  by  ]\Iarlove,  piece  is  found.     The  length  of  the  pipe  from 

the  additional  length  being  302  mm".  (11-9  in.),  the  bottom  of  the  mouth  to  the  open  end  was 

corresponding  exactly  with  half  the  length  of  205  mm.  =8-07  inch  ;  the  node,  as  determmed, 

wave  of  the  pipe.     The  position  of  the  nodal  was  137  mm.  =  5-39  inch  from  the  open  end, 

plane  in  the  inside  of  the  pipe  was  determined  and  G8  mm.  =2-68  inch  from  the  bottom  of  the 

by  inserting  a  wooden  plug  of  the  same  diameter  mouth.     These  lengths  had  to  be  increased  by 

as  of  the  internal  opening  of  tlie  pipe  at  its  14  mm.  =  -55  in.  and  83  mm.  =  3-27  in.  respec- 

open  end,  until  the  pitch    of  the  pipe,  whicli  tively,  to  make  up  each  to  the  quarter  length 

had  now  become  a  closed  one,  was  exactly  the  of  the  wave  151  mm.  =  5-95  inch. —  Translator.^ 
same  as  that  of  the  open  pipe  before  the  inser- 


92  MUSICAL  TONES  OF  FLUTE  OR  FLUE  PIPES.  part  i. 

air  is  blown  out  of  such  a  slit  as  that  which  lies  below  the  lip  of  the  pipe,  it  l)vexks 
through  the  air  which  lies  at  rest  in  front  of  the  slit  in  a  thin  sheet  like  a  blade  or 
lamina,  and  hence  at  first  does  not  draw  any  sensible  part  of  that  air  into  its  own 
motion.  It  is  not  till  it  reaches  a  distance  of  some  centimetres  [a  centimetre  is 
nearly  four-tenths  of  an  inch]  that  the  outpouring  sheet  splits  up  into  eddies  or 
vortices,  which  effect  a  mixture  of  the  air  at  rest  and  the  air  in  motion.  This 
blade-shaped  sheet  of  air  in  motion  can  be  rendered  visible  by  sending  a  stream  of 
air  impregnated  with  smoke  or  clouds  of  salammoniac  through  the  mouth  of  a 
pipe  from  which  the  pipe  itself  is  removed,  such  as  is  commonly  found  among 
physical  apparatus.  Any  blade-shaped  gas  flame  which  comes  from  a  split  burner 
is  also  an  example  of  a  similar  process.  Burning  i-enders  visible  the  limits  between 
the  outpoiu-ing  sheet  of  gas  and  the  atmosphere.  But  the  flame  does  not  render 
the  continuation  of  the  stream  visible. 
^  Now  the  blade-shaped  sheet  of  air  at  the  mouth  of  the  organ  pipe  is  wafted  to 
one  side  or  the  other  by  every  stream  of  air  which  touches  its  surface,  exactly  as 
this  gas  flame  is.  The  consequence  is  that  when  the  oscillation  of  the  mass  of  air 
in  the  pipe*  causes  the  air  to  enter  through  the  ends  of  the  pipe,  the  blade-shaped 
stream  of  air  arising  from  the  mouth  is  also  inclined  inwards,  and  drives  its  whole 
mass  of  air  into  the  pipe.t  During  the  opposite  phase  of  vibration,  on  the  other 
hand,  when  the  air  leaves  the  ends  of  the  pipe  the  whole  mass  of  this  blade  of  air 
is  driven  outwards.  Hence  it  happens  that  exactly  at  the  times  when  the  air  in 
the  pipe  is  most  condensed,  more  air  still  is  driven  in  from  the  bellows,  whence 
the  condensation,  and  consequently  also  the  equivalent  of  work  of  the  vibration  of 
the  air  is  increased,  while  at  the  periods  of  rarefaction  in  the  i)ipe  the  wind  of  the 
bellows  pours  its  mass  of  air  into  the  open  space  in  front  of  the  pipe.  We  must 
remember  also  that  the  l)lade-shaped  sheet  of  air  requires  time  in  order  to  traverse 
the  width  of  the  mouth  of  the  pipe,  and  is  during  this  time  exjjosed  to  the  action 
Ijof  the  vibrating  column  of  air  in  the  pipe,  and  does  not  reach  the  lip  (that  is  tlie 
line  where  the  two  paths,  inwards  and  oiitwards,  intersect)  until  the  end  of  this 
time.  Every  particle  of  air  that  is  blown  in,  consequently  reaches  a  phase  of 
vibration  in  the  interior  of  the  pipe,  which  is  somewhat  later  than  that  to  which 
it  was  exposed  in  traversing  the  opening.  If  the  latter  motion  was  inwards,  it 
encounters  the  following  condensation  in  the  interior  of  the  pipe,  and  so  on. 

This  mode  of  exciting  the  tone  conditions  also  the  peculiar  quality  of  tone  <jf 
these  organ  pipes.  We  may  regard  the  blade-shaped  stream  of  air  as  very  thin  in 
comparison  with  the  amplitude  of  the  vibrations  of  air.  The  latter  often  amount 
to  10  or  16  millimetres  {-39  to  -63  inches),  as  may  be  seen  by  bringing  small 
flames  of  gas  close  to  this  opening.  Consequently  the  alternation  between  the 
periods  of  time  for  which  the  whole  blast  is  poured  into  the  interior  of  the  pipe, 
and  those  for  which  it  is  entirely  emptied  outside,  is  rather  sudden,  in  fact  almost 
instantaneous.  Hence  it  follows  it  that  the  oscillations  excited  by  blowing  are  of 
^  a  similar  kind  ;  namely,  that  for  a  certain  part  of  each  vibration  the  velocity  of  the 
particles  of  air  in  the  mouth  and  in  free  space,  have  a  constant  value  directed  out- 
wards, and  for  a  second  portion  of  the  same,  a  constant  value  directed  inwards. 
With  stronger  blowing  that  directed  inwards  will  be  more  intense  and  of  shorter 
duration  ;  with  weaker  blowing,  the  converse  may  take  place.  Moreover,  the  pres- 
sure in  the  mass  of  air  put  in  motion  in  the  ijipe  must  also  alternate  between  two 
constant  values  with  considerable  rapidity.  The  rapidity  of  this  change  will, 
however,  be  moderated  by  the  circumstance  that  the  blade-shaped  sheet  of  air  is 
not  infinitely  thin,  but  recpiires  a  short  time  to  pass  over  the  lip  of  the  pipe,  and 

*  [It  has,  however,  not  been  explained  bow  side  the  pipe  is  very  small.  A  candle  tlame 
that  '  oscillation '  commences.  This  will  be  held  at  tlie  end  of  the  pipe  only  pulsates ; 
alluded  to  in  the  additions  to  App.  VII.  sect.  B.  held  a  few  inches  from  the  lip,  along  the  edge 
Translator.]  of  the  pipe,  it  is  speedily  extinguished.— 7'*r<?ts- 

t  [The  amount  of  air  which  enters  as  com-  /ahir.'] 
pared  with  that  which  passes  over  the  lip  out-  J  See  Appendix  VII.  [especially  sect.  B,  II.] . 


CHAP.  V.  5.         MUSICAL  TONES  OF  FLUTE  OH  FLUE  I'lUES.  9;J 

that  secondly  the  higher  upper  partials,  whose  wave-lengtlis  only  slightly  exceed  the 
diameter  of  the  pipe,  are  as  a  general  rule  imperfectly  developed. 

The  kind  of  motion  of  the  air  here  described  is  exactly  the  same  as  that  shown 
in  Hg.  23  (p.  82/>),  B  and  C,  fig.  24  (p.  836),  A  and  B,  for  the  vibrating  points  of 
a  violin  string.  Organ-builders  have  long  since  remarked  the  similarity  of  the 
(piality  (jf  tone,  for  the  narrower  cylindrical-pipe  stops  when  strongly  blown,  as 
shown  by  the  names  :   Geigenprindpal ,   Viola  di  Gamha,  Violonvello,   Violon-fHisit* 

That  these  conclusions  from  the  mechanics  of  blowing  correspond  with  the 
facts  in  nature,  is  shown  by  the  experiments  of  Messrs.  Toepler  tt  Boltzmann,t  who 
rendered  the  form  of  the  oscillation  of  pressure  in  the  interior  of  the  pipe  optically 
observable  by  the  interference  of  light  passed  through  a  node  of  the  vibrating  mass 
of  air.  When  the  force  of  the  wind  was  small  they  found  almost  a  simple  vibration 
(the  smaller  the  oscillation  of  the  aii'-blade  at  the  lip,  the  more  completely  the  dis- 
continuities disappear).  But  when  the  force  of  the  wind  was  greater  they  found  H 
a  very  rapid  alternation  between  two  different  values  of  pressure,  each  of  which 
)'emained  almost  unaltered  for  a  fraction  of  a  vibration. 

Messrs.  Mach  and  J.  Hervert's  J  experiments  with  gas  flames  placed  before  the 
end  of  an  open  pipe  to  make  the  vibrations  visible,  show  that  the  form  of  motion 
just  described  really  occurs  at  the  ends  of  the  pipes.  The  forms  of  vibration  which 
they  deduced  from  the  analysis  of  the  forms  of  the  flames  correspond  with  those  of 
a  violin  string,  except  that,  for  the  reason  given  above,  their  corners  are  rounded  off. 
By  using  resonators  I  find  that  on  narrow  pipes  of  this  kind  the  partial  tones 
may  be  clearly  heard  up  to  the  sixth. 

For  wide  open  pipes,  on  the  other  hand,  the  adjacent  proper  tones  of  the  tube 
are  all  somewhat  sharper  than  the  corresponding  iiarmonic  tones  of  the  prime,  and 
hence  these  tones  will  be  much  less  reinforced  by  the  resonance  of  the  tube.  Wide 
pipes,  having  larger  masses  of  vibrating  air  and  admitting  of  being  much  more 
strongly  blown  without  jumping  up  into  an  harmonic,  are  used  for  the  great  body^ 
of  sound  on  the  organ,  and  are  hence  called  p^'in^ipalstimmen.i.  For  the  above 
reasons  they  produce  the  prime  tone  alone  strongly  and  fully,  with  a  much  weaker 
retinue  of  secondary  tones.  For  wooden  'principal'  pipes,  I  find  the  prime  tone 
and  its  Octave  or  first  upper  partial  very  distinct ;  the  Twelfth  or  second  upper 
partial  is  but  weak,  and  the  higher  upper  partials  no  longer  distinctly  perceptible. 
For  metal  pipes  the  fourth  partial  was  also  still  perceptible.  The  quality  of  tone  in 
these  pipes  is  fuller  and  softer  than  that  of  the  geigenjyrincipal*  When  flute  or 
flue  stops  of  the  organ,  and  the  German  flute  are  blown  softly,  the  upper  partials 
lose  strength  at  a  greater  rate  than  the  prime  tone,  and  hence  the  musical  quality 
becomes  weak  and  soft. 

Another  variety  is  observed  on  the  pipes  which  are  conically  narrowed  at  their 

*  [GeiyeiipriHcipal— violin     or     crisp-toned  sively  conical  with  a  bell  top.     From  Hopkins 

diapason,  8  feet, — violin  principal,  4  feet.    See  on  the  0?-gan,  pp.  137,  445,  &c. — Translator.] 
supra,  p.  91rf,  note.     Violoncello — 'crisp-toned  -f  Poggendor&'s  A7inal.,  vol.  cxli.  pp.  321- 

open  stop,  of  small  scale,  the  Octave  to  the  352.  ^ 

violone,    8   feet'.       Violon-hass —thin   fails   in  :f  Poggendorff's ^-i?n!r?/.,  vol.  cxlvii.  pp.  590- 

Hopkins,   but    it   is   probably   his    'violone^  604. 

double  bass,  a  unison  open  wood  stop,  of  much  §  [Literally   '  principal   voices   or    parts  ' ; 

smaller  scale  than  the  Diapason,  and  formed  may  probably   be   best  translated  '  principal 

of  pipes  that  are  a  little  wider  at  the  top  than  work  '  or  '  diapason-work,'  including  '  all  the 

at  the  bottom,  and  furnished  with  ears  and  open    cylindrical    stops    of    Open    Diapason 

beard  at  the  mouth  ;  the  tone  of  the  Violone  measure,  or  which  have  their  scale  deduced 

is  crisp  and  resonant,  like  that  of  the  orches-  from  that  of  the  Open  Diapason ;  such  stops 

tral  Double  Bass  ;  and  its  speech  being  a  little  are  the  chief,  most  important  or  "■  princijiul,'" 

slow,  it  has  the  Stopped  Bass  always  drawn  as  they  are  also  most  numerous  in  an  organ. 

with  it,  16  feet'.     Gamha  or  viol  da  gamba —  The   Unison   and    Double    Open    Diapasons, 

'  l)ass  viol,  unison  stop,  of  smaller  scale,  and  Principal,  Fifteenth   and   Octave   Fifteenth  ; 

thinner  but  more  pungent  tone  than  the  violin  the  Fifth,  Twelfth,  and  Larigot;  the  Tenth 

diapason,  8  feet,   .   .   .  one  of  the  most  highly  and  Tierce  ;  and  the  Mixture  Stops,  when  of 

esteemed  and  most  frequently  disposed  stops  full  or  proportional  scale,  belong  to  the  Dia- 

in  Continental  organs ;  the  German  gamba  is  pason-work.'     From  Hopkins  on  the  Organ, 

usually  composed  of  cylindrical  pipes  '.      In  p.  131. —  Translator  J] 
England  tiU  very  recently  it  was  made  exclu- 


94  MUSICAL  TONES  (3F  FLUTE  OR  FLUE  PIPES.  part  i. 

upper  end,  in  the  mUcwnnl,  f/emshom,  und  spitzfllite  stops.*  Their  upper  opening 
has  generally  half  the  diameter  of  the  lower  section;  for  the  same  length  the 
mlicional  pipe  has  the  narrowest,  and  the  sjnt^ote  the  widest  section.  These  pipes 
have,  I  find,  the  property  of  rendering  some  higher  partial  tones,  from  the  Fifth 
to  the  Seventh,  comparatively  stronger  than  the  lower.  The  tiuality  of  tone  is 
consequently  poor  but  peculiarly  bright. 

The  narrower  stopped  cylindrical  pnpes  have  proper  tones  correspondmg  to  the 
unevenly  numljered  partials  of  the  prime,  that  is,  the  third  partial  or  Twelfth,  the 
fifth  partial  or  higher  major  Third,  and  so  on.  For  the  wider  stopped  pipes,  as  for 
the  wide  open  pipes,  the  next  adjacent  proper  tones  of  the  mass  of  air  are  distinctly 
higher  than  the  corresponding  upper  partials  of  the  prime,  and  consequently  these 
upper  partials  are  very  slightly,  if  at  all,  reinforced.  Hence  wide  stopped  pipes, 
especially  when  gently  blown,   give  the  prime  tone  almost  alone,  and  they  were 

H  therefore  j)reviously  adduced  as  examples  of  simple  tones  (p.  60c).  Narrow  stopped 
pipes,  on  the  other  hand,  let  the  Twelfth  be  very  distinctly  heard  at  the  same 
time  with  the  prime  time  ;  and  have  hence  been  called  quintaten  {<juintam  tenentes).f 
When  these  pipes  are  strongly  blown  they  also  give  the  fifth  partial,  or  higher 
major  Third,  very  distinctly.  Another  variety  of  quality  is  produced  by  the 
rohrfl'ute.X  Here  a  tube,  open  at  both  ends,  is  inserted  in  the  cover  of  a  stopped 
pipe,  and  in  the  examples  I  examined,  its  length  was  that  of  an  open  pipe  giving 
the  fifth  partial  tone  of  the  prime  tone  of  the  stopped  pipe.  T'he  fifth  partial  tone 
is  thus  proportionably  stronger  than  the  rather  weak  third  partial  on  these  pipes, 
and  the  quality  of  tone  becomes  peculiarly  bright.  Compared  with  open  pipes  the 
quality  of  tone  in  stopped  pipes,  where  the  evenly  numbered  partial  tones  are 
absent,  is  somewhat  hollow  ;  the  wider  stopped  i)ipes  have  a  dull  quality  of  tone, 
especially  when  deep,  and  are  soft  and  powerless.  But  their  softness  offers  a  very 
effective  contrast  to  the  more  cutting  qualities  of  the  narrower  open  pipes  and  the 

U  noisy  corni)oiind  stops,  of  which  I  have  already  spoken  (p.  576),  and  which,  as  is 
well  known,  form  a  compound  tone  by  uniting  many  pipes  corresponding  to  a  prime 
and  its  upper  partial  tones. 

Wooden  pipes  do  not  produce  such  a  cutting  windrush  as  metal  pipes.  Wooden 
sides  also  do  not  resist  the  agitation  of  the  waves  of  sound  so  well  as  metal  ones,  and 
hence  the  vibrations  of  higher  pitch  seem  to  be  destroyed  by  friction.  For  these 
reasons  wood  gives  a  softer,  but  duller,  less  penetrating  quality  of  tone  than  metal. 
It  is  characteristic  of  all  pipes  of  this  kind  that  they  speak  readily,  and  hence 
admit  of  great  rapidity  in  musical  divisions  and  figures,  but,  as  a  little  increase  of 
force  in  blowing  distinctly  alters  the  pitch,  their  loudness  of  tone  can  scarcely  be 
changed.  Hence  on  the  ovgan  forte  and  ^r/a«o  have  to  be  produced  by  stops,  which 
regulate  the  introduction  of  pipes  with  various  qualities  of  tone,  sometimes  more, 
sometimes  fewer,  now  the  loud  and  cutting,  now  the  weak  and  soft.  The  means  of 
expression  on  this  instrument  are  therefore  somewhat  limited,  but,  on  the  other 

■^  hand,  it  clearly  owes  part  of  its  magnificent  properties  to  its  power  of  sustaining 
tones  with  unaltered  force,  undisturbed  by  subjective  excitement. 

*  \Saliclonnl—'  reedy  Double  Dulciana,  16  conical  bodies,  8  feet '.     '  This  stop  is  found  of 

feet  and  8  feet,  octave  salicional,  4  feet '.     The  8,  4,  and  2  feet  length  in  German  organs.     In 

Dulciana  is  described  as 'belonging  to  the  Flute-  England  it  has  hitherto  been  made  chiefly  as  a 

work  the  pipes  much  smaller  in  scale  than  4-feet  stop  ;  i.e.  of  principal  pitch.     The  pipes 

those  of  the  open  diapason  .  .  .  tone  peculiarly  of  the  Spitz-flute  are  slightly  conical,  being 

soft  and  gentle  '  (Hopkins,  p.  113).     Gemshorn,  about  J  narrower  at  top  than  at  the  mouth, 

literally  '  chamois  horn ' ;  in  Hopkins,  '  Goat-  and  the  tone  is  therefore  rather  softer  than 

horn,  a  unison  open  metal  stop,  more  conical  that  of  the  cylindrical  stop,  but  of  very  pleas- 

than'  the  Spitz-Flote,  8  feet '.     '  A  member  of  ing  quality '  {ibid.  p.  l-^0).^Translator.'] 
the  Flute-work  and  met  with  of  8,  4,  or  2  feet  t  [See  supra,  p.  33(7,  note.—  Translator.'] 

len<^th  in  Continental  organs.    The  pipes  of  this  \  [Itoh rflote-  '  Double  Stopped  Diapason  of 

stop  are  only  i  the  diameter  at  the  top  that  they  metal  pipes  with  chimneys,  16  feet,  Keed-flute, 

are  at  the  mouth ;  and  the  tone  is  consequently  Metal  Stopped  Diapason,  with  reeds,  tubes  or 

liaht    but   very   clear   and    travelling '    {ibid.  chimneys,  8  feet.     Stopped  ]\Ietal  Flute,  with 

priio).      Spitzflotc—' S^ire  or  taper  flute,  a  reeds,  tubes  or  chimneys,  4  feet'  (Hopkins, 

unison  open  metal  stop  formed  of  pipes  with  pp.  444,  U5).—  Tra)tslator.] 


€HAP.    V.    6. 


Ml'SlCAL  TONES  OF  REEL)   PIPES. 


95 


6.     Mii^irnl    Tniu'!^  nf   Red   Piju-s. 

The  mode  of  producing-  the  tones  on  these  instrnments  resembles  that  used  for 
the  siren  :  the  passage  for  the  air  being  alternately  closed  and  opened,  its  stream  is 
separated  into  a  series  of  individual  pulses.  This  is  effected  on  the  siren,  as  we 
have  already  seen,  by  means  of  a  rotating  disc  pierced  with  holes.  In  reed  instru- 
ments, elastic  plates  or  tongues  are  employed  which  are  set  in  vibration  and  thus 
alternately  close  and  open  the  aperture  in  which  they  are  fastened.  'I'o  tliese 
belong— 

1.  The  reed  pipes  of  organs  mid  the  luhraiors  of  hunDouliiinx.  Their  tongues, 
shown  in  perspective  in   fig.   28,  A,  and  in  section   in   tig.   28,    B,  are  thin  oblong 

metal    plates,   z  z,   fastened 
"^"^  "^'  to    a   brass    block,    a  a,    in 

which  there  is  a  hole,  b  b,  *\ 
behind  the  tongue  and  of 
the  same  shape.  When  the 
tongue  is  in  its  position  of 
rest,  it  closes  the  hole  com- 
pletely, with  the  exception 
of  a  very  fine  chink  all  round 
'^       ;'...  '^~— ^  "  its  margin.    When  in  motion 

it  oscillates  between  the  po- 
sitions marked  Zj  and  z.,  in  tig.  28,  B.  In  the  position  Zi  there  is  an  aperture  for 
tlie  stream  of  air  to  enter,  in  the  direction  shown  by  the  arrow,  and  this  is  closed 
when  the  tongue  has  reached  the  other  extreme  position  z,,.  The  tongue  shown 
is  a  free  vibrator  or  anche  lihre,  such  as  is  now  universally  employed.  These 
tongues  are  slightly  smaller  than  the  corresponding  opening,  so  that  they  can  bend 
inwards  without  touching  the  edges  of  the  hole.*  Formerly,  striking  vibrators^ 
■or  reeds  were  employed,  which  on  each  oscillation  struck  against  their  frame. 
But  as  these  produced  a  harsh  quality  of  tone  and  an  uncertain  pitch  they  have 
gone  out  of  use.* 


fa/- 


*  [The  quaUty  of  tone  produced  by  the  free 
reed  can  be  greatly  modified  by  comparatively 
slight  changes.  If  the  reed  is  quite  flat,  the 
end  not  turning  up,  as  it  does  in  fig.  28,  above, 
no  tone  can  be  produced.  If  the  size  of  the 
slit  round  the  edges  be  enlarged,  by  forcing  a 
thin  plate  of  steel  between  tlie  spring  and  the 
flange,  and  tlien  withdrawing  it,  the  quality  of 
tone  is  permanently  changed.  Another  change 
is  produced  by  curving  the  middle  part  up  and 
then  down  in  a  curve  of  contrary  flexure. 
Another  change  results  from  curving  the  ends 
of  the  reed  up  as  in  '  American  organs ' — a 
species  of  harmonium.  One  of  the  earliest  free 
reed  instruments  is  tlie  Chinese  '  sheng,'  which 
Mr.  Hermann  Smith  thus  describes  from  his 
ownspecimen.  Seealso App. XX.sect.K.  'The 
body  of  the  instrument  is  in  the  form  and  size 
of  a  teacup  with  a  tightly  fitting  cover,  pierced 
with  a  series  of  lioles,  arranged  in  a  circle,  to 
receive  a  set  of  small  pipe-like  canes,  17  in 
number,  and  of  various  lengths,  of  which  13 
are  capable  of  sounding  and  4  are  mute,  but 
necessary  for  structure.  The  lower  end  of  each 
pipe  is  fitted  with  a  little  free  reed  of  very 
delicate  workmanship,  about  half  an  inch  long, 
and  stamped  in  a  thin  metal  plate,  liaving  its 
tip  slightly  loaded  with  beeswax,  whicb  is  also 
used  for  keeping  the  reed  in  position.  One 
peculiarity  to  be  noticed  is  that  the  reed  is 
quite  level  with  the  face  of  the  plate,  a  condi- 
tion in  which   modern   free  reeds  would  not 


speak.  But  this  singular  provision  is  made  to 
ensure  speaking  either  by  blowing  or  suction. 
The  corners  of  the  reeds  are  rounded  off,  and 
thus  a  little  space  is  left  between  the  tip  of  the 
reed  and  the  frame  for  the  passage  of  air,  an 
arrangement  quite  adverse  to  the  speaking  of 
harmonium  reeds.  In  each  pipe  the  integrity 
of  the  column  of  air  is  broken  by  a  hole  in 
the  side,  a  short  distance  above  the  cup.  By 
this  strange  contrivance  not  a  single  pipe  will 
sound  to  the  wind  blown  into  the  cup  from 
a  flexible  tube,  until  its  side  hole  has  been 
covered  by  the  finger  of  the  player,  and  then 
the  pipe  gives  a  note  corresponding  to  its  full  * 
speaking  length.  Wliatever  be  the  speaking 
length  of  the  pipe  the  hole  is  placed  at  a  short 
distance  above  the  cup.  Its  position  has  no 
relation  to  nodal  distance,  and  it  effects  its 
purpose  by  breaking  up  the  air  column  and 
preventing  it  from  furnishing  a  proper  recipro- 
cating relation  to  the  pitch  of  the  reed.'  The 
instrument  thus  described  is  the  '  sing '  of 
Barrow  (IVavels  in  China,  1804,  where  it  is 
well  figured  as  '  a  pipe,  with  unequal  reeds 
or  bamboos'),  and  '  le  petit  cheug '  of  Pere 
Amiot  [M&inoircs  concernant  Vhistoirc  .  .  ■ 
dcs  Chinois,  .  .  .  1780,  vol.  vi.,  where  a  'cheng  ' 
of  24  pipes  is  figured. — Translator.'] 

t  [It  will  be  seen  by  App.  VII.  to  this 
edition,  end  of  sect.  A.,  that  Prof.  Helmholtz 
has  somewhat  modified  his  opinion  on  this 
point,  in   consequence   of   the   information  I 


90 


MUSICAL  TONES  OF  REED  PIPES. 


'J'lie  mode  in 
ill  tig.  29,  A  ai 
dinal    section 
into    which    the    wind 
tonijue  1  is    fastened 


which  tongues  are  fastened 
id   B  below 
p  ])  is   the 


A   bears   a 
air   chamber 
is   driven  ;     the 
1    tlie    groove  r, 


in  the  reed  stops  of 
sonant    cnj)    above ; 

A  F[fi.  20. 


organs 
B    is   a 


is  sliown 
longitu- 


which  fits  into  the  block  s ;  d  is  the 
tuning  wire,  which  presses  against  the 
tongue,  and  being  pushed  down  shortens 
it  and  hence  sharpens  its  pitch,  and, 
conversely,  flattens  tlie  pitch  when  pulled 
up.  Slight  variations  of  pitch  are  thus 
easily  produced.* 

2.  The  tongues  of  clarinets,  oboes,  and 
^\  bassoons,  are  constructed  in  a  somewhat 
similar  manner  and  are  cut  out  of  elastic 
reed  plates.  The  clarinet  has  a  single 
wide  tongue  which  is  fastened  before  the 
corresponding  opening  of  the  mouth- 
piece like  the  metal  tongues  previous^ 
described,  and  would  strike  the  frame  if 
its  excursions  were  long  enough.     But  its 


obtained  from  some  of  the  principal  English 
organ- builders,  which  I  here  insert  from  p.  711 
of  the  first  edition  of  this  translation : — Mr. 
Willis  tells  nie  that  he  never  uses  free  reeds, 
that  no  power  can  be  got  from  them,  and  that 
he  looks  upon  them  as  '  artificial  toys '. 
Messrs.  J.  W.  Walker  &  Sons  say  that  they 

^  have  also  never  used  free  reeds  for  the  forty  or 
more  years  that  they  have  been  in  business, 
and  consider  that  free  reeds  have  been  super- 
seded by  striking  reeds.  Mr.  Thomas  Hill 
informs  me  that  free  reeds  had  been  tried  by 
his  father,  by  M.  Cavaille-Coll  of  Paris,  and 
others,  in  every  imaginable  way,  for  the  last 
thirty  or  forty  years,  and  were  abandoned  as 
'  utterly  worthless  '.  But  he  mentions  that 
Schiilze  (of  Paulenzelle,  Schwartzburg)  told 
him  that  he  never  saw  a  striking  reed  till 
he  came  over  to  England  in  1851,  and  that 
Walcker  (of  Ludwigsburg,  Wuertemberg)  had 
Jittle  experience  of  theni,  as  Mr.  Hill  learnt 
from  him  aljout  twenty  years  ago.  j\Ir.  Hill 
adds,  however,  that  both  these  builders  speedily 
abandoned  the  free  reed,  after  seeing  the 
English  practice  of  voicing  striking  reeds. 
This  is  corroborated  by  Mr.  Hermann  Smith's 

^  statement  (1875)  that  Schulze,  in  1862,  built 
the  great  organ  at  Doncaster  with  94  stops, 
of  which  only  the  Trombone  and  its  Octave 
had  free  reeds  (see  Hopkins  on  the  Organ, 
p.  530,  for  an  account  of  this  organ) ;  and 
that  two  years  ago  he  built  an  organ  of  64 
stops  and  4,052  pipes  for  Sheffield,  with  not 
one  free  reed ;  also  that  Walcker  built  the 
great  organ  for  Ulm  cathedral,  with  6,500 
pipes  and  100  stops,  of  which  .34  had  reeds, 
and  out  of  them  only  2  had  free  reeds ;  and 
that  more  recently  he  built  as  large  a  one  for 
pjoston  ]\Iusic  Hall,  without  more  free  reeds  ; 
and  again  that  CavailM-CoU  quite  recently 
built  an  organ  for  Mr.  Hopwood  of  Kensington 
of  2,252  pipes  and  40  stops,  of  which  only  one 
— the  Musette — had  free  reeds.  He  also  says 
that  Lewis,  and  probably  most  of  the  London 
organ-builders  not  previously  mentioned,  have 
never  used  the  free  reed.    The  harshness  of  the 


striking  reed  is  obviated  in  the  English  method 
of  voicing,  according  to  Mr.  H.  Smith,  by  so 
curving  and  manipulating  the  metal  tongue, 
tliat  instead  of  coming  with  a  discontinuous 
'  flap  '  from  the  fixed  extremity  down  on  to  the 
slit  of  the  tube,  it  '  rolls  itself '  down,  and 
hence  gradually  covers  the  aperture.  The  art 
of  curving  the  tongue  so  as  to  produce  this 
effect  is  very  difficult  to  acquire  ;  it  is  entirely 
empirical,  and  depends  upon  the  keen  eye  and 
fine  touch  of  the  '  artist,'  who  notes  lines  and 
curves  imperceptible  to  the  uninitiated  obser- 
ver, and  foresees  their  influence  on  the  produc- 
tion of  quality  of  tone.  Consequently,  when  an 
organ-builder  has  the  misfortune  to  lose  his 
'  reed-voicer,'  he  has  always  great  difficulty 
in  replacing  him. —  Translator.'] 

"  [It  should  be  observed  that  fig.  29,  A, 
sliows  a /rcc  reed,  and  fig.  29,  B,  a  strikivg  reed ; 
and  that  the  tuning  wire  is  right  in  fig.  29,  B, 
because  it  presses  the  reed  against  the  edges  of 
its  groove  and  hence  shortens  it,  but  it  is  wrong 
in  fig.  29,  A,  for  the  reed  being  free  would  strike 
against  the  wire  and  rattle.  For  free  reeds  a 
clip  is  used  which  grasps  the  reed  on  both  sides 
and  thus  limits  its  vibrating  length. 

Fig.  28,  p.  9bh,  shows  the  vibrator  of  an 
harmonium ,  not  of  an  organ  pipe.  The  figures 
are  the  same  as  in  all  the  German  editions. — 
Translator.] 


6. 


TONES  OF  KEED  PIPES. 


excursions  arc  small,  and  the  pressure  of  the  lips  hriuys  it  just  near  enough  to 
make  the  chink  sufficiently  small  without  allowing  it  to  strike.  For  the  oboe  and 
bassoon  two  reeds  or  tongues  of  the  same  kind  are  placed  opposite  each  other  at  the 
end  of  the  mouthpiece.  They  are  separated  by  a  narrow  chink,  and  by  blowing  are 
pressed  near  enough  to  close  the  chink  whenever  they  swing  inwards. 

3.  Memhmnoiis  t(»i(/tt('s. — The  peculiarities  of  these  tongues  are   best    studied 
on  those  artificially  constructed.     Cut   the  end  of  a  wooden  or  gutta-percha  tube 
Pjp  ..„  obliquely  on  both  sides,   as   shown  in  fig.   30, 

leaving  two  nearly  rectangular  points  standing 
between  the  two  edges  which  are  cut  obliquely. 
Then  gently  stretch  strips  of  vulcanised  India 
rubber  over  the  two  oblique  edges,  so  as  to  leave 
a  small  slit  between  them,  and  fasten  them  with 
a  thread.  A  reed  mouthpiece  is  thus  constructed  1^ 
which  may  be  connected  in  any  way  with  tubes 
or  other  air  chambers.  When  the  membranes 
V  bend  inwards  the  slit  is  closed;  when  outwards, 

^  it  is  open.     Membranes  which  are  fastened  in 

this  oblique  manner  speak  much  better  than  those  which  are  laid  at  right  angles 
to  the  axis  of  the  tube,  as  Johannes  Miiller  proposed,  for  in  the  latter  case  they 
require  to  be  bent  outwards  by  the  air  before  they  can  begin  to  open  and  shut 
alternately.  Membranous  tongues  of  the  kind  proposed  may  be  blown  either  in 
the  direction  of  the  arrows  or  in  the  opposite  direction.  In  the  first  case  they  open 
the  slit  when  they  move  towards  the  air  chamber,  that  is,  towards  the  further  end 
of  the  conducting  tube.  Tongues  of  this  kind  I  distinguish  as  striking  inwards. 
When  blown  they  always  give  deeper  tones  than  they  would  do  if  allowed  to 
vibrate  freely,  that  is,  without  being  connected  with  an  air  chamber.  The  tongues 
of  organ  pipes,  harmoniums*,  and  wooden  wind  instruments  already  mentioned,  ^ 
are  likewise  always  arranged  to  strike  inwards.  But  both  membranous  and  metal 
tongues  may  be  arranged  so  as  to  act  against  the  stream  of  air,  and  hence  to  open 
when  they  move  towards  the  outer  opening  of  the  instrument.  I  then  say  that  they 
strike  ovfrnmls.  The  tones  of  tongues  which  strike  outwards  are  always  sharper 
than  those  of  isolated  tongues. 

Only  two  kinds  of  membranous  tongues  liave  to  be  considered  as  musical  in- 
struments :  the  human  lips  in  brass  instruments,  and  the  human  lari/n.r  in  sim/in;/. 
The  lips  must  be  considered  as  very  slightly  elastic  membranous  tongues, 
loaded  with  much  inelastic  tissue  containing  water,  and  they  would  consequently 
vibrate  very  slowly,  if  they  could  be  brought  to  vibrate  by  themselves.  In  brass 
instruments  they  form  membranous  tongues  which  strike  outwards,  and  conse- 
cpiently  bythe  above  rule  produce  tones  sharper  than  their  proper  tones.  But  as 
they  offer  very  slight  resistance,  they  are  readily  set  in  motion,  by  the  alternate 
pressure  of  the  vibrating  coliunn  of  air,  when  used  with  brass  instruments.*  ^ 


*  '"Mr.  D.  J.  Blaikley  (manager  of  Messrs. 
Boosey  &  Co.'s  Military  Musical  Instrument 
Manufactory,  who  has  studied  all  such  instru- 
ments theoretically  as  well  as  practically,  and 
read  many  papers  upon  them,  to  some  of  which 
I  shall  have  to  refer)  finds  that  this  statement 
does  not  represent  his  own  sensations  when 
playing  the  horn.  '  The  lips,'  he  says,  '  do  not 
vibrate  througliout  their  whole  length,  but  only 
through  a  certain  length  determined  by  the 
diameter  of  the  cup  of  the  mouthpiece.  Pro- 
bably also  the  vibrating  length  can  be  modified 
by  the  mere  pinch,  at  least  this  is  the  sensa- 
tion I  experiencedwhen  sounding  high  notes  on  a 
large  mouthpiece.  The  compass  (about  4  octaves) 
possible  on  a  given  mouthpiece  is  much  greater 
than  that  of  any  one  register  of  the  voice,  and 


the  whole  range  of  brass  instruments  played 
thus  with  the  lips  is  about  one  octave  greater 
than  the  whole  range  of  the  human  voice  from 
basso  prof  undo  to  the  highest  soprano.  That 
the  lips,  acting  as  the  vocal  chords  do,  can 
themselves  vibrate  rapidly  when  supported  by 
the  rim  of  a  mouthpiece,  may  be  proved,  for  if 
such  a  rim,  unconnected  with  any  resonating 
tube,  be  held  against  the  lips,  various  notes  of 
the  scale  can  be  produced  very  faintly,  the  dif- 
ficulty being  to  maintain  steadiness  of  pitch 
{Fhilos.  May.,  Aug.  1878,  p.  2).  TU  office  of 
the  air  in  the  tube  in  relation  to  the  lips  (leav- 
ing out  of  consideration  its  work  as  a  resonant 
body,  intensifying  and  modifying  the  tone)  is 
to  ad  as  a  pendulum  governor  in  facilitating 
the   maintenance   (not   the    origination)    of   a 

H 


98  TONES  OF  HEED  PIPES.  part  i. 

In  the  larynx,  the  ehistic  vocal  cliords  act  as  membranous  tongues.  They  arc 
stretched  across  the  windpipe,  from  front  to  back,  like  the  india-rubber  strips  in 
fig.  30  (p.  97a),  and  leave  a  small  slit,  the  glottis,  between  them.  They  have  the 
advantage  over  all  artificially  constructed  tongues  of  allowing  the  width  of  their  slit, 
their  tension,  and  even  their  form  to  be  altered  at  pleasure  with  extraordinary 
rapidity  and  certainty,  at  the  same  time  that  the  resonant  tube  formed  l)y  the 
opening  of  the  mouth  admits  of  much  variety  of  form,  so  that  many  more  qualities 
of  tone  can  be  thus  produced  than  on  any  instrument  of  artificial  construction.  If 
the  vocal  chords  are  examined  from  above  with  a  laryngoscope,  while  producing  a 
tone,  they  will  be  seen  to  make  very  large  vibrations  for  the  deeper  breast  voice, 
shutting  the  glottis  tightly  whenever  they  strike  inwards. 

The  pitch  of  the  various  reeds  or  tongues  just  mentioned  is  altered  in  very 
different  manners.       The  metal  tongues   of  the   organ  and  harmonium  are  always 

^  intended  to  produce  one  single  tone  apiece.  On  the  motion  of  these  comparatively 
heav}'  and  stiff  tongues,  the  pressure  of  the  vibrating  air  has  very  small  influence, 
and  their  pitch  within  the  instrument  is  consequently  not  much  different  from  that 
of  the  isolated  tongues.  There  must  be  at  least  one  tongue  for  each  note  on  such 
instruments. 

In  wooden  witul  instruments,  a  single  tongue  has  to  serve  for  the  whole  series 
of  notes.  But  the  tongues  of  these  instruments  are  made  of  light  elastic  wood, 
which  is  easily  set  in  motion  by  the  alternating  pressure  of  the  vibrating  column 
of  air,  and  swings  sympathetically  with  it.  Such  instruments,  therefore,  in 
addition  to  those  very  high  tones,  which  nearly  correspond  to  the  proper  tones  of 
their  tongues,  can,  as  theory  and  experience  alike  show,  also  produce  deep  tones  of 
a  very  different  pitch,*  because  the  waves  of  air  which  arise  in  the  tube  of  the  in- 
strument excite  an  alternation  in  the  pressure  of  air  adjacent  to  the  tongue  itself 
sufficiently  powerful  to  make   it  vibrate  sensibly.     Now  in  a  vibrating  column   of 

H  air  the  alteration  of  pressure  is  greatest  where  the  velocity  of  the  particles  of  air  is 
smallest ;  and  since  the  velocity  is  always  null,  that  is  a  minimum,  at  the  end  of  a 
closed  tube,  such  as  a  stopped  organ  pipe,  and  the  alteration  of  pressure  in  that 
place  is  consequently  a  maximum,  the  tones  of  these  reed  pipes  must  be  the  same  as 
those  which  the  resonant  tube  alone  would  produce,  if  it  were  stopped  at  the  place 
where  the  tongue  is  placed,  and  were  blown  as  a  stopped  pipe.  In  musical  practice, 
then,  such  tones  of  the  instrument  as  correspond  to  the  proper  tones  of  the  tongue 
are  not  used  at  all,  because  they  are  very  high  and  screaming,  and  their  pitch  can- 
not be  preserved  with  sufficient  steadiness  when  the  tongue  is  wet.  The  only 
tones  produced  are  considerably  deeper  than  the  proper  tone  of  the  tongue,  and 
have  their  pitches  determined  by  the  length  of  the  column  of  air,  which  corresponds 
to  the  proper  tones  of  the  stopped  pipe. 

The  clarinet  has  a  cylindrical  tube,  the  proper  tones  of  which  correspond  to 
the  third,  fifth,  seventh,  ifec,  partial  tone   of   the  prime.     By  altering  the  style  of 

^i  blowing,  it  is  possible  to  pass  from  the  prime  to  the  Twelfth  or  the  higher  major 
Third.     The  acoustic  length  of  the  tube  may  also  be  altered  by  opening  the  side 

periodic  vibration  of  the  lips.     Prof.  Helmholtz  which  he  produced  a  tone  of  40  vib.,  the  tone 

does  not  say  above  what  produces  the  alternate  was,  even  at  that  depth,  remarkably  rich  and 

pressure,  and  I  can  conceive  no  source  for  it  but  fine,  owing  to  the  large  and  deep  cup  extinguish- 

a  periodic  vibration  of  the  lips  of  a  time  suited  ing  the  beating  upper  partials.     ]\Ir.  Blaikley 

to  the  particular  note  required.'  The  depth  of  also  drew  my  attention  to  the  fact  that  where 

thecupisalsoimportant:— 'The  shallower  and  the  tube  opens  out  into  the  cup,  there  must 

more  "cup-like"  the  cup,'  says  Mr.  Blaikley,  be  no  sharp  shoulder,  but  that  the  edge  must 

'  the  greater  the  strength  of  the  upper  partials.  be  carefully  rounded  off,  otherwise  there  is  a 

Compare   the   deep   and   narrow   cup   of   the  great  loss  of  power  to  the  blower.     In  the  case 

French  horn  with  weak  upper  partials,  and  of  the  Frencli  horn  the  cup  is  very  long  and 

the  wide  and  shallow  cup  of  the  trmnpet  with  almost  tapers  into  the  tube. — Translator.] 
strongupperpartials.'— (MS. communications.)  *  See   Helmholtz,    Verhandluiujen  dcs   tm- 

]\Ir.  Blaikley  kindly  sounded  for  me  the  same  turhistorischen   medicinischen    Vereins  zu   Hei- 

instrument  with  different  mouthpieces  or  cups,  delbercj.     July   26,    1861,   in   the    Hcidelbergcr 

to  show  the  great  difference  of  quality  they  Jahrbilcher.       Poggendorff's     Annalen,     1861. 

produce.      In  the  great  bass  bombard'^on  on  [Reproduced  in  part  in  App.  VII.  sect.  B.,  I.] 


CHAP.  V.   6. 


TONES  OF  REED  PIPES. 


on 


holes  of  the  clarinet,  in  which  case  the  vibrating  column  of  air  is  principally  that 
between  the  mouthpiece  and  the  uppermost  open  side  hole.* 

The  oho(^  (hautbois)  and  hassoon  (fagotto)  have  conical  tubes  which  are  closed  up 
to  the  vertex  of  their  cone,  and  have  proper  tones  that  arc  the  same  as  those  of 
open  tubes  of  the  same  length.  Hence  the  tones  of  both  of  these  instniments 
nearly  correspond  to  those  of  open  pipes.  By  overblowing  they  give  the  Octave, 
Twelfth,  second  Octave,  and  so  on,  of  the  prime  tone.  Intermediate  tones  are 
produced  by  opening  side  holes. 

The  older  horns  and  trunipeis  consist  of  long  conical  bent  tubes,  witliout  keys 
or  side  holes. t  They  can  produce  such  tones  only  as  correspond  to  the  proper 
tones  of  the  tube,  and  these  again  are  the  natural  harmonic  upper  partials  of  the 
prime.  But  as  the  prime  tone  of  such  a  long  tube  is  very  deep,  the  upper  pai'tial 
tones  in  the  middle  parts  of  the  scale  lie  rather  close  together,  especially  in  the 
extremely  long  tubes  of  the  horn,J  so  that  they  give  most  of  the  degrees  of  the  scale.  ^ 


*  [Tdr.D.  J.  Blaikley  obligingly  furnished  me 
with  the  substance  of  the  following  remarks  on 
clarinets,  and  repeated  his  experiments  before 
me  in  May  1884.  The  ordinary  form  of  the 
clarinet  is  not  wholly  cylindrical.  It  is  slightly 
constricted  at  the  mouthpiece  and  provided 
with  a  spreading  bell  at  the  other  end.  The 
modification  of  form  by  key  and  finger  holes 
also  must  not  be  neglected.  On  a  cylindrical 
pipe  played  with  the  lips,  the  evenly  numbered 
partials  are  quite  inaudible.  When  a  clarinet 
mouthpiece  was  added  I  found  traces  of  the 
4th  and  6th  partials  beating  with  my  forks. 
But  on  the  clarinet  with  the  bell,  the  2nd, 
4th,  and  6th  partials  were  distinct,  and  I  could 
obtain  beats  from  them  with  my  forks.  ]Mr. 
Blaikley  brought  them  out  (1)  by  bead  and 
diaphragm  resonators  tuued  to  them  (fig.  15, 
p.  42rt),  which  I  also  witnessed,  (2)  by  an  irre- 
gularly-shaped tubular  resonator  sunk  gra- 
dually in  water,  on  which  I  also  heard  them. 


(3)  by  beats  with  an  harmonium  with  a  con- 
stant blast,  which  I  also  heard.  On  the  cylin- 
drical tube  all  the  unevenly  numbered  partials 
are  in  tune  when  played  as  primes  of  inde- 
pendent harmonic  notes.  On  the  clarinet 
only  the  3rd  partial,  or  2nd  proper  tone,  can 
be  used  as  the  prime  of  an  independent  har- 
monic tone.  The  3rd,  4th,  and  5th  proper 
tones  of  the  instrument,  are  sufficiently  near 
in  pitch  to  the  5th,  7th,  and  9th  partials  of 
the  fundamental  tone  for  these  latter  to  be 
greatly  strengthened  by  resonance,  but  the 
agreement  is  not  close  enough  to  allow  of  the 
higher  proper  tones  being  used  as  the  primes 
of  independent  harmonic  compound  tones. 
Hence  practically  only  the  3rd  harmonics, 
or  Twelfths,  are  used  on  the  clarinet.  The 
following  table  of  the  relative  intensity  of  the  H 
partials  of  a  B\)  clarinet  was  given  by  Mr. 
Blaikley  in  the  Proc.  of  the  Mus.  Assn.  for 
1877-8,  p.  84  :— 

Partials— B|j  Clarinets. 


Pitch 

1 

•2 

3 

4 

5 

(3 

7 

8,  &c. 

/' 

/■ 

rS 

,/• 

P 

mf 

P 

... 

^b 

,/ 

^ 

f 

V 

mf 

mf 

PP 

t, 

T 

s 

r 

P 

vif 

mf 

t'P 

<i 

T 

X 

f 

nif 

mf 

P 

PP 

f 

f 

-« 

f 

mf 

P 

mf 

PP 

i 

f 

"S 

mf 

P 

P 

mf 

PP 

d 

J 

t-5 

mf 

P 

■mf 

P 

P 

PP 

Where  /■  means  forte,  ;«/ mezzoforte 

t  [Such  brass  tubes  are  first  worked  unbent 
from  cylindrical  brass  tubes,  by  putting  solid 
steel  cores  of  the  required  form  inside,  and  then 
drawing  them  through  a  hole  in  a  piece  of 
lead,  which  yields  enough  for  the  tube  to  pass 
through,  but  presses  the  brass  firmly  enough 
against  the  core  to  make  the  tube  assume  the 
proper  form.  Afterwards  the  tube  is  filled 
with  lead,  and  then  bent  into  the  required  coils, 
after  which  the  lead  is  melted  out.  The  in- 
struments are  also  not  conical  in  the  strict 
sense  of  the  word,  but  '  approximate  in  form 
to  the  hyperbolic  cone,  where  the  axis  of  the 
instrument  is  an  asymptote,  and  the  vertex  is 
at  a  great  or  even  an  infinite  distance  from 
the  bell  end  '.  From  information  furnished  by 
Mr.  B\-A\k\ey.— Translator.  ] 

I  The  tube  of  the  Wahlhoni  [foresthorn. 
Notes         .         .         .      e'\y       f  (f 

Just  cents.  .  .  0,  204,  386, 
Harmonic  cents  .  0,  204,  386, 
Harmonics,  No.         .       8,         9,  10, 


jo  piano,  yp  pianissimo.     Translator.] 

hunting  horn  of  the  Germans,  answering  to 
our  French  horn]  is,  according  to  Zamminer 
[p.  312],  13-4  feet  long.  Its  proper  prime  tone  ff 
is  E\\).  This  and  the  next  £J'rf  are  not  used, 
but  only  the  other  tones,  B\f,  e\y,  g,  b\f,  rf'jj  -  , 
e'jj,  /',  r/,  a'\f+,  V\y,  &c.  [Mr.  Blaikley 
kindly  sounded  for  me  the  harmonics  8,  9,  10, 
11,  12,  13,  14  on  an  E,\)  French  horn.  The 
result  was  almost  precisely  320,  360,  400,  440, 
480,  520,  560  vib.,  that  is  the  exact  harmonics 
for  the  prime  tone  40  vib.  to  which  it  was 
tuned,  the  pitch  of  English  military  musical 
instruments  being  as  nearly  as  possible  c  269, 
e'\)  319-9,  a'  452-4.  This  scale  was  not  com- 
pleted because  the  15th  and  IGth  harmonics 
600  and  640  vib.  would  have  been  too  high  for 
me  to  measure.  Expressed  in  cents  we  may 
compare  this  scale  with  just  intonation  thus : — 

«'b       ^'b       «"       'i"\)       d"         «"b 

498,       702,       884,       996,       1088,       1200 

551,       702,       841,       969,       1088,       1200 

11,         12,         13,         14,  15,  16. 

H  2 


100  TONES  OF  REED  PIPES.  pakt  i. 

The  trumpet  is  restricted  to  these  natural  tones.  But  by  introducing  tlie  hand 
into  the  bell  of  the  French  horn  and  thus  partly  closing  it,  and  by  lengthening 
the  tube  of  the  trombone,*  it  was  possible  in  some  degree  to  siipj)ly  the  missing 
tones  and  improve  the  faulty  ones.  In  later  times  trumpets  and  horns  have  been 
frequently  supplied  with  keys  t  to  supply  the  missing  tones,  but  at  some  expense 
of  power  in  the  tone  and  the  brilliancy  in  its  quality.  The  vibrations  of  the  air 
in  these  instruments  are  unusually  powerful,  and  require  the  I'esistance  of  firm, 
smooth,  unbroken  tubes  to  preserve  their  strength.  In  the  use  of  brass  instru- 
ments, the  different  form  and  tension  of  the  lips  of  the  player  act  only  to  determine 
which  of  the  proper  tones  of  the  tube  shall  speak  ;  the  pitch  of  the  individual 
tones  is  almost  J  entirely  independent  of  the  tension  of  the  lips. 

On  the  other  hand,  in  the  /Ki\//n.r  the  tension  of  the  vocal  chords,  which  here 
form   the   membranous   tongues,   is    itself   variable,    and    determines   the    pitch    of 

^the  tone.  The  air  chambers  connected  with  the  lai'vnx  are  not  adapted  for 
materially  altering  the  tone  of  the  vocal  chords.  Their  walls  are  so  yielding  that 
they  cannot  allow  the  formation  of  vibrations  of  the  air  within  them  sufficiently 
powerful  to  force  the  vocal  chords  to  oscillate  with  a  period  which  is  diftcrent  from 
that  required  by  their  own  elasticity.  The  cavity  of  the  mouth  is  also  far  too 
short,  and  generally  too  widely  open  for  its  mass  of  air  to  have  material  influence 
on  the  pitch. 

In  addition  to  the  tension  of  the  vocal  chords  (which  can  be  increased  not 
only  by  separating  the  points  of  their  insertion  in  the  cartilages  of  the  larynx,  but 
also  by  voluntarily  stretching  the  muscular  fibres  within  them),  their  thickness 
seems  also  to  be  variable.  Much  soft  Avatery  inelastic  tissue  lies  luiderneath  the 
elastic  fibrils  proper  and  the  nuiscular  fibres  of  the  vocal  chords,  and  in  the  breast 
voice  this  probably  acts  to  weight  them  and  retard  their  vibrations.  The  head 
voice  is  probably  produced  by  drawing  aside  the  mucous  coat  below  the  chords, 

H  thus  rendering  the  edge  of  the  chords  sharper,  and  the  weiglit  of  the  vibrating 
part  less,  wdiile  the  elasticity  is  unaltered,  j 

Hence  the  Fourth  a"f^  was  53  cents  (33  :  32)  trombone  can  be  altered  at  will,  and  cliosen 

too   sharp,  and   the    Sixth   c"  was   43   cents  to  make  its  harmonics  produce  a  just  scale. 

(40  :  39)  too  flat,  and  they  were  consequently  Some  trumpets  also  are  made  with  a  short 

unusable  without  modification  by  the  hand.  slide   worked   by   two   fingers   one   way,   and 

Themiuor  Seventh  (^'[j  was  too  flat  by  27  cents  returning  to  its  position  by  a  spring.     Such 

(64  :  G3),   but    unless    played    in    (intended)  instruments  are  sometimes  used  by  first-rate 

unison  against  the  just  form,  it  produces  a  players,  such  as  Harper,  the  late  celebrated 

better  effect.     'In  trumpets,  strictly  so  called,'  trumpeter,  and  his  son.     But,  as  Mr.  Blaikley 

says  Mr.  Blaikley, '  a  great  portion  of  the  length  informed  me,  an  extremely  small  percentage 

is  cylindrical  and  the  bell  curves  out  hyper-  of  the  trumpets  sold  have  slides.     At  present 

bolically,    the    two    lowest    partials    are   not  the  piston  brass  instruments  have  nearly  driven 

required   as   a   rule   and   are   not   strictly  in  all  slides,  except  the  trombone,  out  of  the  field, 

tune,  so  the  series  of  partials  may  be  taken  — Translatnr.'] 

as  about  -75,  1-90,  3,  4,  5,  6,  7,  8,  &c.,  all  the  f  [The  keys  are  nearly  obsolete,  and  have 

upper  notes  being  brought  into  tune  by  modi-  been  replaced  by  pistons  which  open  valves, 

ITficationsin  theformof  the  bellinagoodinstru-  and  thus  temporarily  increase  the  length  of 

ment.'     The  length  of  the  French  horn  varies  the  tube,  so  as  to  make  the  note  blown  1,  2, 

with  the  '  crook '  which  determines  its  pitch.  or  3  Semitones  flatter.      These  can  also  be 

The  following  contains  the  length  in  English  used  in  combination,  but  are  then  not  so  true, 

inches  for  each  crook,  as  given  by  Mr.  Blaiklev :  This   is   tantamount   to    an    imperfect    slide 

B\)    (alto)  108,  ^h  1141,  ^j^  121J,  G  128f,>  action.      Instruments  of  this   kind   are  now 

1441,  ^|5   i53_  E\y  162,  Z»i3  171J,  C  192f,  B\f  much   used   in   all   military   bands,  and   are 

(basso)     216J,  hence   the   length  varies  from  made  of  very  different  sizes  and   pitches. — 

9  ft.  to  18  ft.  f  inch.     By  a  curious  error  in  Translator.'] 

all  the  German  editions,  Zamminer  is  said  to  I  [But  by  no  means  '  quite  '.     It  is  possible 

make  the  length  of  the  E\y  Waldhorn  27  feet,  to  blow  out  of  tune,  and  to  a  small  extent 

or  the  length  of  the  wave  of  the  loiccd  note,  temper  the  harmonics. — Translator.'] 
in  place  of  his  13"4  feet.     Zamminer,  however,  §  [On  the  suljject  of  the  registers  of   tlie 

says  that  the  instrument  is  named  from  the  human  voice  and  its  production  generally,  see 

Octave  ahuve  the  lowest  note,  and  that  hence  Lennox  Browne  and  Emit  Behnke,  Voice,  ^ong, 

the  wave-length  of  this  Octave  is  the  length  of  and    Speech    (Sampson    Low,    London,    1883, 

the  horn. —  Translator.]  pp.    322).       This   work   contains   not   merely 

*  [A  large  portion  of  the  trombone  is  com-  accui'ate  drawings  of  the  larynx  in  the  different 

posed  of  a  double  narrow  cylindrical  tube  on  registers,    but    4    laryngoscopic    photographs 

which  another  slides,  so  that  the  length  of  the  from  Mr.   Behnke's   own  larynx.      A  reijist''r 


TONES  OF  r»KED  ril'KS' 


■OT 


We  now  proceed  to  investigate  the  '/ua/if//  <>/  tn)i,-  prudueed  on  reed  |)ii)es, 
which  is  our  proper  subject.  The  sound  in  these  pipes  is  excited  by  intermittent 
pulses  of  air,  which  at  each  swing  break  through  the  opening  that  is  closed  by 
the  tongue  of  the  reed.  A  freely  vibrating  tongue  has  far  too  small  a  surface  to 
communicate  any  appreciable  qiiantity  of  sonorous  motion  to  the  surrounding  air ; 
and  it  is  as  little  able  to  excite  the  air  inclosed  in  pipes.  The  sound  seems  to  be 
re;dly  produced  by  pulses  of  air,  as  in  the  siren,  where  the  metal  plate  that  opens 
and  closes  the  oriiice  does  not  vibrate  at  all.  By  the  alternate  opening  and  closing 
of  a  passage,  a  continuous  influx  of  air  is  changed  into  a  periodic  motion,  capable 
of  affecting  the  air.  Like  any  other  periodic  motion  of  the  air,  the  one  thus 
produced  can  also  be  resolved  into  a  seines  of  simple  vibrations.  We  have  already 
remarked  that  the  number  of  terms  in  such  a  series  will  increase  with  the  discon- 
tinuity of  the  motion  to  be  thus  resolved  (p.  ?>\d).  Now  the  motion  of  the  air  which 
passes  through  a  siren,  or  past  a  vibrating  tongue,  is  discontinuous  in  a  very  high  H 
degree,  since  the  individual  pulses  of  air  must  be  generally  separated  by  complete 
pauses  during  the  closures  of  the  opening.  Free  tongues  without  a  resonance 
tube,  in  which  all  the  individual  simple  tones  of  the  vibration  which  they  excite 
in  the  air  are  given  oft"  freely  to  the  surrounding  atmosphere,  have  consequently 
always  a  very  sharp,  cutting,  jarring  quality  of  tone,  and  we  can  really  hear  with 
either  armed  or  unarmed  ears  a  long  series  of  strong  and  clear  partial  tones  up 
to  the  l(3th  or  20th,  and  there  are  evidently  still  higher  partials  present,  although 
it  is  difficult  or  impossible  to  distinguish  them  from  each  other,  because  they  do 
not  lie  so  nuich  as  a  Semitone  apart.*  This  whirring  of  dissonant  partial  tones 
makes  the  musical  quality  of  free  tongues  very  disagreeable. t  A  tone  thus  pro- 
duced also  shows  that  it  is  really  due  to  puffs  of  air.  I  have  examined  the  vibra- 
ting tongue  of  a  reed  pipe,  like  that  in  fig.  28  (p.  95/y),  when  in  action  with  the 
vibration  microscope  of.  Lissajous,  in  order  to  determine  the  vibrational  form  of 
the  tongue,  and  I  found  that  the  tongue  performed  perfectly  regular  simple  vibra-  ^ 
tions.  Hence  it  would  communicate  to  the  air  merely  a  simple  tone  and  not  a 
compound  tone,  if  the  sound  were  directly  produced  by  its  own  vibrations. 

The  intensity  of  the  upper  partial  tones  of  a  free  tongtie,  unconnected  with  a 
resonance  tube,  and    their   relation  to    the    prime,  are   greatly  dependent  on  the 


is  defined  as  '  a  series  of  tones  produced  by 
the  same  mechanism  '  (p.  163).  The  names  of 
the  registers  adopted  are  those  introduced 
by  the  late  John  Curwen  of  the  Tonic  Sol-fa 
movement.  They  depend  on  the  appearance  of 
the  glottis  and  vocal  chords,  and  are  as  follows  : 
1.  Lower  thick,  2.  Upper  thick  (both  '  chest 
voice  ■),  3.  Lower  thin  ('high  chest'  voice  in 
men),  4.  Upper  thin  ('falsetto'  in  women), 
.5.  Small  ('  head  voice '  in  women).  The  extent 
of  the  registers  are  stated  to  be  (p.  171) 

1.  lower  tliick.  2.  upper  thick.  3.  lower  thin. 
/Men          Eioa,           //to/',  ,'/'  to  ,•" 

(Women   c  toe',  (/'to/,  /toe" 

1.  lower  thick.  -L  upper  thick,  o.  lower  tliiu. 


Women  only. 


«/"  to/",         g"  to/' 
4.  upper  thin.        5.  small. 


The  mechanism  is  as  follows  (pp.  163-171)  : — 

1.  Lower  thick.  The  hindmost  points  of  the 
pyramids  (arytenoid  cartilages)  close  together, 
an  elliptical  slit  between  the  vocal  ligaments 
(or  chords),  which  vibrate  through  their  whole 
length,  breadth,  and  thickness  fully,  loosely, 
and  visibly.     The  lid  (epiglottis)  is  low. 

2.  Upper  thick.  "The  elliptical  chink  dis- 
appears and  becomes  linear.  The  lid  (epiglottis) 
rises  ;  the  vocal  ligaments  are  stretched. 

3.  Lower  thin.  The  lid  (epiglottis)  is  more 
raised,  so  as  to  show  the  cushion  below  it,  the 
whole  larynx  and  the  insertions  of  the  vocal 


ligaments  in  the  shield  (thyroid)  cartilage. 
The  vocal  ligaments  are  quite  still,  and  their 
vibrations  are  confined  to  the  thin  inner  edges. 
The  vocal  ligaments  are  made  thinurr  and 
transparent,  as  shown  by  illumination  from 
lielow.     Male  voices  cease  here. 

4.  Upper  thin.  An  elliptical  slit  again  forms 
between  the  vocal  ligaments.  When  this  is 
used  by  men  it  gives  the  falsetto  arising  from 
the  upper  thin  being  carried  below  its  true 
place.  This  slit  is  gradually  reduced  in  size 
as  the  contralto  and  soprano  voices  ascend.       U, 

5.  Small.  The  back  part  of  the  glottis 
contracts  for  at  least  two-thirds  of  its  length, 
the  vocal  ligaments  being  pressed  together  so 
tightly  that  scarcely  any  trace  of  a  slit  remains, 
and  no  vibrations  are  visible.  The  front  part 
opens  as  an  oval  chink,  and  the  edges  of  this 
vibrate  so  markedly  that  the  outline  is  blurred. 
The  drawings  of  the  two  lost  registers  (pp.  168- 
169)  were  made  from  laryugoscopic  examina- 
tion of  a  lady. 

Keference  should  be  made  to  the  book 
itself  for  full  explanations,  and  the  reader 
should  especially  consult  Mr.  Behnke's  admir- 
able little  work  The  Mechanism  of  the  H^iman 
rokc  (Curwen,  3rd  ed.,  1881,  pp.  125).— 
Trauslatur.] 

*  [See  footnote  t  p.  b(kl' .  —  Translator.] 
■\-  [The    cheap   little   mouth   harmonicons 
exhibit  this  effect  very  well. —  Translofor.] 


10:^  TONES  OF  KEED  PIPES.  paht  i. 

nature  of  the  tongue,  its  position  with  respect  tt)  its  frame,  the  tightness  with 
whi(;h  it  closes,  &c.  Striking  tongues  which  produce  the  most  discontinuous  pulses 
of  air,  also  produce  the  most  cutting  quality  of  tone.*  The  shorter  the  puff  of  air, 
and  hence  the  more  sudden  its  action,  the  greater  number  of  high  upper  jjartials 
should  we  expect,  exactly  as  we  find  in  the  siren,  according  to  Seebeck's  investi- 
gations. Hard,  unyielding  material,  like  that  of  brass  tongues,  will  produce 
pulses  of  air  which  are  much  more  disconnected  than  those  formed  by  soft  and 
yielding  substances.  This  is  probably  the  reason  why  the  singing  tones  of  the 
human  voice  are  softer  than  all  others  which  are  produced  by  reed  pipes.  Never- 
theless the  number  of  upper  partial  tones  in  the  human  voice,  when  used  in 
emphatic  foj-fe,  is  very  great,  and  they  reach  distinctly  and  powerfully  up  to  the 
four-times  accented  [or  quarter-foot]  Octave  (p.  26*^/).  To  this  we  shall  have  to 
return . 
^  The  tones  of  tongues  are  essentially  changed  by  the  addition  of  resonance 
tubes,  because  they  reinforce  and  hence  give  prominence  to  those  upper  partial 
tones  which  correspond  to  the  proper  tones  of  these  tubes.t  In  this  case  the 
resonance  tubes  must  be  considered  as  closed  at  the  point  where  tlie  tongue  is 
inserted.! 

A  brass  tongvie  such  as  is  used  in  organs,  and  tuned  to  /y[7,  was  applied  to  one 
of  my  larger  spherical  resonators,  also  tuned  to  fj\},  instead  of  to  its  usual  resonance 
tube.  After  considerably  increasing  the  pressure  of  wind  in  the  bellows,  the 
tongue  spoke  somewhat  flatter  than  usual,  but  with  an  extraordinarily  full,  beautiful, 
soft  tone,  from  which  almost  all  upper  partials  were  absent.  Very  little  wind  was 
used,  but  it  was  under  high  pressure.  In  this  case  the  prime  tone  of  the  compound 
was  in  unison  with  the  resonator,  which  gave  a  powerful  resonance,  and  conse- 
quently the  prime  tone  had  also  great  power.  None  of  the  higher  partial  tones 
could  be  reinforced.  The  theory  of  the  vibrations  of  air  in  the  sphere  further 
U  show^s  that  the  greatest  pressure  must  occur  in  the  sphere  at  the  moment  that  the 
tongue  opens.  Hence  arose  the  necessity  of  strong  pressure  in  the  bellows  to  over- 
come the  increased  pressure  in  the  sphere,  and  yet  not  much  wind  really  passed. 

If  instead  of  a  glass  sphere,  resonant  tubes  are  employed,  which  admit  of  a 
greater  number  of  proper  tones,  the  resulting  musical  tones  are  more  complex. 
In  the  clarinet  w^e  have  a  cylindrical  tube  which  by  its  resonance  reinforces  the 
uneven  partial  tones.§  The  conical  tubes  of  the  oboe,  bassoon,  trimipet,  and 
French  horn,  on  the  other  hand,  reinforce  all  the  harmonic  upper  partial  tones  of 
the  compound  up  to  a  certain  height,  determined  by  the  incapacity  of  the  tubes 
to  resound  for  waves  of  sound  that  are  not  much  longer  than  the  width  of  the 
opening.  By  actiud  trial  I  found  only  unevenly  numbered  partial  tones,  distinct  to 
the  seventh  inclusive,  in  the  notes  of  the  clarinet,^  whereas  on  other  instmmients, 
wdiich  have  conical  tubes,  I  found  the  evenly  numbered  partials  also.  I  have  not  yet 
had  an  opportunity  of  making  observations  on  the  further  differences  of  quality  in 
f  the  tones  of  individual  instruments  with  conical  tubes.  This  opens  rather  a  wide 
field  for  research,  since  the  quality  of  tone  is  altered  in  many  ways  by  the  style  of 
blowing,  and  even  on  the  same  instrument  the  different  parts  of  the  scale,  when 
they  require  the  opening  of  side  holes,  show  considerable  differences  in  quality. 
On  wooden  wind  instruments  these  differences  are  striking.  The  opening  of  side 
holes  is  by  no  means  a  complete  substitute  for  shortening  the  tube,  and  the  reflec- 
tion of  the  waves  of  sound  at  the  i>oints  of  opening  is  not  the  same  as  at  the  free 
open  end  of  the  tube.  The  upper  partials  of  compound  tones  produced  by  a  tube 
limited  by  an  open  side  hole,  must  certainly  be  in  general  materially  deficient  in 
harmonic  purity,  and  this  will  also  have  a  marked  influence  on  their  resonance.** 

*  [But    see    footnote    f    P-    95(>' .  — Trans-  p.    89,    1.    2,    but   was   cancelled    in    the   4th 

lator.]  German  edition.  —  Trctiisthitor.] 

+  [A  line  lias  been   here  cancelled  in  the  I   See  Appendix  VII. 

translation  which  had  been  accidentally  left  §  [But  see  note  *  p.  d'db.— Translator.] 

standing  in  the  German,  as  it  refers  to  a  re-  **  [The  theory  of  side  holes  is  excessively 

mark  on  the  passage  which  formerly  followed  complicated  and  has  not  been  as  yet  worked 


CHAP.  V.  7.  VOWEL  QUALITIKS  OF  TONE.  lO'A 

7.    Voi'rl  Qualities  of  Torn'. 

We  have  hitherto  discussed  cases  of  resonance,  generated  in  such  air  chambers 
as  were  capable  of  reinforcing  the  prime  tone  principally,  but  also  a  certain 
number  of  the  harmonic  upper  partial  tones  of  the  compound  tone  produced.  The 
case,  however,  may  also  occur  in  which  the  lowest  tone  of  the  resonance  chamber 
applied  does  not  con-espond  with  the  prime,  but  only  with  some  one  of  the  upper 
partials  of  the  compound  tone  itself,  and  in  these  cases  we  find,  in  accordance  with 
the  principles  hitherto  developed,  that  the  corresponding  upper  partial  tone  is 
really  more  reinforced  than  the  pi'ime  or  other  partials  by  the  resonance  of  the 
chamber,  and  consequently  predominates  extremely  over  all  the  other  partials  in 
the  series.  The  quality  of  tone  thus  produced  has  consequently  a  peculiar  cha- 
racter, and  more  or  less  resembles  one  of  the  vowels  of  the  human  voice.  For  the 
vowels  of  speech  are  in  reality  tones  produced  by  membi'anous  tongues  (the  vocal  "i 
chords),  with  a  resonance  chamber  (the  mouth)  capable  of  altering  in  length, 
width,  and  pitch  of  resonance,  and  hence  capable  also  of  reinforcing  at  difterent 
times  difl'erent  partials  of  the  compound  tone  to  which  it  is  aiiplied.* 

In  order  to  understand  the  composition  of  vowel  tones,  we  must  in  the  first 
place  bear  in  mind  that  the  source  of  their  sound  lies  in  the  vocal  chords,  and 
that  when  the  voice  is  heard,  these  chords  act  as  membranous  tongues,  and  like 
all  tongues  produce  a  series  of  decidedly  discontinuous  and  sharply  separated 
pulses  of  air,  which,  on  being  represented  as  a  sum  of  simple  vibrations,  must 
consist  of  a  very  large  number  of  them,  and  hence  be  received  by  the  ear  as  a  very 
long  series  of  partials  belonging  to  a  compoiuid  musical  tone.  With  the  assistant 
of  resonators  it  is  possible  to  recognise  very  high  partials,  up  to  the  sixteenth, 
when  one  of  the  brighter  vowels  is  sung  by  a  powerful  bass  voice  at  a  low  pitch, 
and,  in  the  case  of  a  strained  forte  in  the  upper  notes  of  any  human  voice,  we  can 
hear,  more  clearly  than  on  any  other  musical  instnunent,  those  high  upi)er  partials  ^f 
that  belong  to  the  middle  of  the  four-times  accented  Octave  (the  highest  on 
modern  pianofortes,  see  note,  p.  IScZ),  and  these  high  tones  have  a  peculiar  relation 
to  the  ear,  to  be  subsequently  considered.  The  loudness  of  such  upper  partials, 
especially  those  of  highest  pitch,  differs  considerably  in  different  individuals.  For 
cutting  bright  voices  it  is  greater  than  for  soft  and  dull  ones.  The  quality  of  tone 
in  cutting  screaming  voices  may  perhaps  be  referred  to  a  want  of  suflicient 
smoothness  or  straightness  in  the  edges  of  the  vocal  chords,  to  enable  them  to 
close  in  a  straight  narrow  slit  without  striking  one  another.  This  circumstance 
would  give  the  larynx  more  the  character  of  striking  tongues,  and  the  latter  have 
a  much  more  cutting  quality  than  the  free  tongues  of  the  normal  vocal  chords. 
Hoarseness  in  voices  may  arise  from  the  glottis  not  entirely  closing  during  the 
vibrations  of  the  vocal  chords.  At  any  rate,  when  alterations  of  this  kind  are 
made  in  artificial  membranous  tongues,  similar  results  ensue.  For  a  strong  and 
yet  soft  quality  of  voice  it  is  necessary  that  the  vocal  chords  should,  even  when  *i 
most  strongly  vibrating,  join  rectilinearly  at  the  moment  of  approach  with  perfect 
tightness,  effectually  closing  the  glottis  for  the  moment,  but  without  overlapping 

out  scientifically.  '  The  general  principles,'  edited  with  additional  letters  by  W.  S.  Broad- 
writes  Islv.  Blaikley,  '  are  not  difficult  of  com-  wood,  and  published  by  Rudall,  Carte,  &  Co., 
prehension ;  the  difficulty  is  to  determine  quan-  makers  of  bis  flutes.  See  also  Victor  Mabillon, 
titatively  tbe  values  in  each  particular  case.'  Etude  sur  Ic  doUjti  dc  la  FlMe  Boehm,  1882, 
The  paper  by  SchafhJiutl  (writing  under  tlie  and  a  paper  by  M.  Aristide  Cavaillo-Coll,  in 
name  of  Pellisov),  'Tbeorie  gedeckter  cylin-  i:  Echo  Mmicaliov  11  Ja.u.lQm.--Translator. 
drischer  und  coni.scher  Pfeifen  uud  der  Quer-  *  The  theory  of  vowel  tones  was  first  enun- 
fiuten,'  Scbweiger,  Journ.  Ixviii.  183.3,  is  dis-  elated  by  Wheatstone  in  a  criticism,  unfortn- 
figured  by  misprints  so  that  the  formulae  are  nately  little  known,  on  Willis's  experunents. 
unintelligible,  and  the  theory  is  also  extremely  The  latter  are  described  in  the  Traiisadions- 
bazardous.  But  they  are  the  only  papers  I  of  the  Cambridge  Philosophkal  Socicti/,  vol. 
have  found,  and  are  referred  to  by  Theobald  iii.  p.  281,  and  Poggendorff's  Annalcn  der 
Boehm,  Ueber  den  Fldtcnl>an,  Mainz,  1847.  Phi/sik;  vol.  xxiv.  p.  397.  Wheatstone's  rc- 
An  English  version  of  this,  by  himself,  made  port  upon  them  is  contained  in  the  London 
for   Mr.    Rudall   in    1847,   has  recently   been  aiul  H'estniinsler  Eevieio  tov  Octohev  1831. 


104  VOWEL  QUALITIES  OE  TONE.  part  i. 

or  striking  against  each  other.  If  they  do  not  close  perfectly,  the  stream  of  air 
will  not  be  completely  interrnptecl,  and  the  tone  cannot  be  powerful.  If  they 
overlap,  the  tone  must  be  cutting,  as  before  remarked,  as  those  arising  from 
.striking  tongues.  On  examining  the  vocal  chords  in  action  by  means  of  a 
laryngoscope,  it  is  marvellous  to  observe  the  accuracy  with  which  they  close  even 
when  making  vibrations  occupying  nearly  the  entire  breadth  of  the  chords  them- 
(selves.* 

There  is  also  a  certain  difterence  in  the  way  of  putting  on  the  voice  in  speak- 
ing and  in  singing,  which  gives  the  speaking  voice  a  much  more  cutting  quality 
of  tone,  especially  in  the  open  vowels,  and  occasions  a  sensation  of  much  greater 
pi'essure  in  the  larynx.  I  suspect  that  in  speaking  the  vocal  chords  act  as  striking 
tongues,  t 

When  the  mucous  membrane  of  the  larynx  is  affected  with  catarrh,  the 
^  laryngoscope  sometimes  shows  little  flakes  of  nnicus  in  the  glottis.  When  these 
are  too  gi'eat  they  disturb  the  motion  of  the  vibrating  chords  and  make  them  irre- 
gular, causing  the  tone  to  become  unequal,  jarring,  or  hoarse.  It  is,  however,  re- 
markable what  comparatively  large  flakes  of  mucus  may  lie  in  the  glottis  without 
jn-oducing  a  very  striking  deterioration  in  the  quality  of  tone. 

It  has  already  been  mentioned  that  it  is  generally  more  difficult  for  the  un- 
assisted ear  to  recognise  the  upper  partials  in  the  human  voice,  than  in  the  tones 
of  musical  instruments.  Resonators  are  more  necessary  for  this  examination 
than  for  the  analysis  of  any  other  kind  of  musictil  tone.  The  upper  partials  of  the 
human  voice  have  nevertheless  been  heard  at  times  by  attentive  observers.  Rameau 
had  heard  them  at  the  beginning  of  last  century.  And  at  a  later  period  Seller  of 
Leipzig  relates  that  while  listening  to  the  chant  of  the  watchman  during  a  sleepless 
night,  he  occasionall}-  heard  at  first,  when  the  watchman  was  at  a  distance,  the 
Twelfth  of  the  melody,  and  afterwards  the  prime  tone.  The  reason  of  this  difficulty 
H  is  most  probably  that  we  have  all  our  lives  remarked  and  observed  the  tones  of 
the  human  voice  more  than  any  other,  and  always  with  the  sole  object  of  grasping 
it  as  a  whole  and  obtaining  a  clear  knowledge  and  percej^tion  of  its  manifold  changes 
of  quality. 

We  may  certainly  assume  that  in  the  tones  of  the  human  larynx,  as  in  all 
other  reed  instruments,  the  upper  partial  tones  would  decrease  in  force  as  they 
increase  in  pitch,  if  they  could  be  observed  without  the  resonance  of  the  cavity  of 
the  mouth.  In  reality  they  satisfy  this  assumption  tolerably  v/ell,  for  those  vowels 
which  are  spoken  with  a  wide  funnel-shaped  cavity  of  the  mouth,  as  A  [n  in  art],  or 
A  [a  in  hat  lengthened,  which  is  nearly  the  same  as  a  in  have].  But  this  relation  is 
materially  altered  by  the  resonance  which  takes  place  in  the  cavity  of  the  mouth. 
The  more  this  cavity  is  narrowed,  either  by  the  lips  or  the  tongue,  the  more  dis- 
tinctly marked  is  its  resonance  for  tones  of  determinate  pitch,  and  the  more  there- 
fore does  this  resonance  reinforce  those  partials  in  the  compoiind  tone  produced  by 
*Ti  the  vocal  chords,  which  approach  the  favoured  pitch,  and  the  more,  on  the  contrary, 
will  the  others  be  damped.  Hence  on  investigating  the  compound  tones  of  the 
human  voice  by  means  of  resonators,  we  find  pretty  imiforinly  that  the  first  six  to 
eight  partials  are  clearly  perceptible,  but  with  very  different  degrees  of  force  accord- 
ing to  the  different  forms  of  the  cavity  of  the  mouth,  sometimes  screaming  loudly 
into  the  eai',  at  others  scarcely  audible. 

Under  these  circumstances  the  investigation  of  the  resonance  of  the  cavity  of 
the  mouth  is  of  great  importance.  The  easiest  and  surest  method  of  finding  the 
tones  to  which  the  air  in  the  oral  cavity  is  tuned  for  the  different  shapes  it  assumes 

*  [Probably  these  observations  were  made  tiThe    Ciermaii    liabit   of  begimiing  open 

on  the    '  upper  thick  '   register,    because   the  vowels   with  the    '  check  '   or  Arabic    hamza, 

chords  are  then  more  visible.     It  is  evident  which  is  very  marked,  and  instantly  cliarac- 

that  these  theories  do  not  apply  to  the  lower  terises   his  nationality,   is   probal)ly   what   is 

thick,   upper  thin,  and  small   registers,    and  here  alluded  to,  as  occasioning  a  sensation  of 

scarcely  to  the  lower  thin,  as  described  above,  much  greater  pressure.     This  does  not  apply 

footnote,  p.  lOlc.  —  Translator.]  in  the  least  to  English  speakers. — Translator.] 


CHAr.   V.    I. 


VOWEL  (QUALITIES  OF  TONK 


105 


ill  tlie  production  of  vowels,  is  that  which  is  used  for  glass  bottles  uiul  other  spaces 
tilled  with  air.  That  is,  tuning-forks  of  different  pitches  have  to  be  struck  and 
held  before  the  opening  of  the  air  chamber — in  the  present  case  the  open  mouth 
— and  the  louder  the  proper  tone  of  the  fork  is  heard,  the  nearer  does  it  corre- 
spond with  one  of  the  proper  tones  of  the  included  mass  of  air."''"  Since  the  shape 
of  the  oral  cavity  can  be  altered  at  pleasure,  it  can  always  be  made  to  suit  the 
tone  of  any  given  tuning-fork,  and  we  thus  easily  discover  what  shape  the  month 
must  assume  for  its  included  mass  of  air  to  be  tuned  to  a  determinate  pitch. 

Having  a  series  of  tuning-forks  at  command,  I  was  thus  able  to  obtain  tlie 
following  results  : — 

The  pitch  of  strongest  resonance  of  the  oral  cavity  de[)eiids  solely  upon  the 
vowel  for  pronouncing  which  the  mouth  has  been  arranged,  and  alters  considerably 
Um-  even  slight  alterations  in  the  vowel  rpiality,  such,  for  example,  as  occur  in  the 
different  dialects  of  the  same  language.  On  the  other  hand,  the  proper  tones  of  II 
the  cavity  of  the  mouth  are  nearly  independent  of  age  and  sex.  I  have  in  general 
found  the  same  resonances  in  men,  women,  and  children.  The  want  of  space  in 
the  oral  cavity  of  women  and  children  can  be  easily  replaced  by  a  great  closure  of  its 
op'Cning,  which  will  make  the  resonance  as  deep  as  in  the  larger  oral  cavities  of  men.f 

The  vowels  can  be  arranged  in  three  series,  according  to  the  position  of  the 
parts  of  the  mouth,  which  nuiy  be  written  thus,  in  accordance  with  Du  13ois- 
lleymond  the  elder  j" : — 

I 


The  vowel  A   [a  in  father,  or  Scotch  a  in  ijian]  forms  the  common  origin  of 
all  three   series.      With  this  vowel  corresponds  a  funnel-shaped  resonance  cavity,  II 


*  [See  note  *  p.  876,  on  determining  violin 
resonance.  One  difficulty  in  the  case  of  the 
mouth  is  that  there  is  a  constant  tendency  to 
vary  the  shape  of  the  oral  cavity.  Another,  as 
shown  at  the  end  of  the  note  cited,  is  that 
the  same  irregular  cavity,  such  as  that  of  the 
mouth,  often  more  or  less  reinforces  a  large 
number  of  different  tones.  As  it  was  impor- 
tant for  my  phonetic  researches,  I  have  made 
many  attempts  to  determine  my  own  vowel 
resonances,  but  have  hitherto  failed  in  all  my 
attempts. — Translator.] 

t  [Easily  tried  by  more  or  less  covering 
the  top  of  a  tumbler  with  the  hand,  till  it 
resounds  to  any  fork  from  c'  to  d"  or  higher. 
—  Translator.] 

t  Norddcutschc  Zeitschrift,  edited  by  de 
la  Motte  Fouque,  1812.  Kculmus  oder  allgc- 
mcine  Alphahetik,  von  F.  H.  du  Bois-Reymond, 
Berlin,  1862,  p.  152.  [This  is  the  arrange- 
ment usually  adopted.  But  in  1867  Mr. 
IMelville  Bell,  an  orthoepical  teacher  of  many 
years'  standing,  who  had  been  led  profession- 
ally to  pay  great  attention  to  the  shapes  of  the 
mouth  necessary  to  produce  certain  sounds,  in 
his  Visible  Speech ;  the  Science  <>f  Unicersal 
Alphahetics  (London :  Simpkin,  JNIarshall  & 
Co.,  4to. ,  pp.  X.  126,  with  sixteen  lithographic 
tables),  proposed  a  more  elaborate  method  of 
classifying  vowels  by  the  shape  of  the  mouth. 
He  commenced  with  9  positions  of  the  tongue, 
consisting  of  .3  in  which  the  middle,  or  as  he 
terms  it,  '  front '  of  the  tongue  was  raised, 
highest  for  ca  in  seat,  not  so  high  for  a  in  sate, 
and  lowest  for  a  in  sat ;  3  others  in  which  the 
back,  instead   of   the   middle,    of   the   tongue 


was  raised,  highest  for  oo  in  snood,  lower  for  o 
in  node,  and  lowest  for  aw  in  imawed  (none  of 
which  three  are  determined  by  the  position  of 
the  tongue  alone),  and  3  intermediate  positions, 
where  the  whole  tongue  is  raised  almost  evenly 
at  three  different  elevations.  These  9  lingual 
positions  might  be  accompanied  with  the 
ordinary  or  with  increased  distension  of  the 
pharynx,  giving  9  primary  and  9  '  wide " 
vowels.  And  each  of  the  18  vowels,  thus 
produced,  could  be  '  rounded,'  that  is,  modified 
by  shading  the  mouth  in  various  degrees  with 
the  lips.  He  thus  obtains  36  distinct  vowel 
cavities,  among  which  almost  all  those  used 
for  vowel  qualities  in  different  nations  may  be 
placed.  Subsequent  research  has  shown  how 
to  extend  this  arrangement  materially.  See  *\ 
my  Eccrly  English  Pronunciation,  part  iv., 
1874,  p.  1279.  Also  see  generally  my  Pro- 
nunciation for  Siiiijers  (Curwen,  1877,  pp.  246) 
and  Speech  in  Son;/  (Novello,  1878,  pp.  140). 
German  vowels  differ  materially  in  quality 
from  the  English,  and  consequently  complete 
agreement  between  Prof.  Helmholtz's  obser- 
vations and  those  of  any  Englishman,  who 
repeats  his  experiments,  must  not  be  expected. 
I  have  consequently  thought  it  better  in  this 
place  to  leave  his  German  notation  untrans- 
lated, and  merely  subjoin  in  parentheses  the 
nearest  English  sounds.  For  the  table  in  the 
text  we  may  assume  A  to  =  a  in  father,  or  else 
Scotch  a  hi  man  (different  sounds),  E  to  =  c  in 
there,  Ito  =  i  in  nuwhinc,  0  to  =  o  in  more,  U 
to  =  u  in  sure;  and  0  to  =  eii  in  French  «c(« 
or  else  in  peuple  (different  sounds),  and  U  to 
=  n  in  French  pu.— Translator.] 


10()  VOWEL  QUALITIKS  OF  TONE.  part  i. 

enlar-ing  witli  tolera])le  uniformity  from  the  larynx  to  the  lipw.  For  the  vowels  of 
the  lower  series,  0  [o  in  motr]  and  U  [oo  in  2>oor],  the  opening  of  the  mouth  is 
contracted  by  means  of  the  lips,  more  for  U  than  for  O,  while  the  cavity  is  enlarged 
as  much  as  possible  by  depression  of  the  tongue,  so  that  on  the  whole  it  becomes 
like  a  bottle  without  a  neck,  with  rather  a  narrow  mouth,  and  a  single  unbroken 
cavity.*  The  pitch  of  such  a  l)ottle-shaped  chamber  is  lower  the  larger  its  cavity 
and  the  narrower  its  mouth.  Usually  only  one  upper  partial  with  strong  resonance 
can  be  clearly  recognised  ;  when  other  proper  tones  exist  they  are  comparatively 
very  high,  or  have  only  weak  resonance.  In  conformity  witli  these  results,  obtained 
with  glass  bottles,  we  find  that  for  a  very  deep  hollow  U  [oo  in  j^oor  nearly],  where 
the  oral  ca\it\-  is  widest  and  the  mouth  narrowest,  the  resonance  is  deepest  and 
answers  to  the  unaccented  /'.  On  passing  from  U  to  0  [o  in  moir  nearly]  the 
resonance   gradually  rises  ;  and   for  a  full,  ringing,  pure   O   the  pitch  is  ^'j?.      The 

5]  position  of  the  mouth  for  O  is  peculiarly  favourable  for  resonance,  the  opening  of  the 
month  being  neither  too  large  nor  too  small,  and  the  internal  cavity  sufficiently 
spacious.  Hence  if  a  //[?  tiuiing-fork  be  struck  and  held  before  the  mouth  while  O 
is  gently  uttered,  or  the  O-position  merely  assumed  without  really  sjieaking,  the  tone 
of  the  fork  will  resound  so  fully  and  loudly  that  a  large  audience  can  hear  it.  Tlie 
usual  '('  tuning-fork  of  musicians  may  also  be  used  for  this  purpose,  but  then  it  will  be 
necessary  to  make  a  somewhat  duller  0,  if  we  wish  to  bring  out  the  full  resonance. 

On  gradually  bringing  the  shape  of  the  mouth  from  the  position  proper  to  0, 
through  those  due  to  0"  [nearly  o  in  cot,  with  rather  more  of  the  0  sound],  and  A"" 
[nearly  <iu  in  raxi/hf,  with  rather  more  of  the  A  sound]  into  that  for  A  [Scotch  a 
in  man,  with  rather  more  of  an  0  quality  in  it  than  English  a  in  fatho-],  the 
resonance  gradually  rises  an  Octave,  and  reaches  //'[?.  This  tone  corresponds  with 
the  North  German  A:  the  somewhat  brighter  A  [(t  m  fftt/ier]  of  the  English  and 
Italians,  rises  up  to  J"',  or  a  major  Third  higher.      It  is  jjarticularly  remarkable  what 

^  little  differences  in  pitch  correspond  to  very  sensible  varieties  of  vowel  quality  in 
the  neighbourhood  of  A  ;  and  I  should  therefore  recommend  ])hilologists  who  wish 
to  define  the  vowels  of  diffl-i-ent  languages  to  rix  them  by  the  ]jitch  of  loudest 
resonance.! 

For  the  vowels  already  mentioned  I  have  not  been  able  to  detect  any  second 
proper  tone,  and  the  analogy  of  the  phenomena  presented  by  artificial  resonance 
chambers  of  similar  shapes  would  hardly  lead  us  to  expect  any  of  sensible  loudness. 


*  iThis  depix'ssed  position  of  the  tongue  able  to  discrimiimte  vowel  sounds,  is  frequently 
ansYv-ers  better  for  English  rnc  in  saw  than  for  not  acute  for  differences  of  pitch.  The  deter- 
eitbcr  (/  in  moir  or  oo  in  -jwor.  For  the  o  the  mination  of  tbe  pitch  even  under  favour- 
tongue  is  slightlv  more  raised,  especially  at  the  able  circumstances  is  not  easy,  especially,  as  it 
back,  while  for  English  ao  the  back  of  the  will  be  seen,  for  the  higher  pitches.  Without 
tongue  is  almost  as  high  as  for  k,  and  greatly  mechanical  appliances  even  good  ears  are 
impedes  the  oral  cavity.  If,  however,  the  deceived  in  the  Octave.  The  differences  cf 
tongue  be  kept  in  the  position  for  «m;  by  sound-  pitch  noted  by  Helmholtz,  Bonders.  Merkel, 
f|  ing  this  vowel,. and,  while  sounding  it  steadily.  and  Koenig,  as  given  on  p.  109(^  probably  point 
the  lips  be  gradually  contracted,  the  sound  to  fundamental  differences  of  pronunciation, 
will  be  found  to  pass  through  certain  obscure  and  show  the  desirability  of  a  very  extensive 
qualities  of  tone  till  it  suddenly  comes  out  series  of  experiments  being  carried  out  with 
clearly  as  a  sound  a  little  more  like  aw  than  o  special  apparatus,  by  an  operator  with  an 
in  more  (really  the  Danish  aa),  and  then  again  extremely  acute  musical  ear,  on  speakers  of 
passing  tbroiigh  other  obscure  phases,  comes  various  nationalities  and  also  on  various 
out  again  clearlv  as  a  deep  sound,  not  so  bright  speakers  of  the  same  nationality.  Great  diffi- 
as  our  (.(1  in  /'om;  but  more  resembhng  the  culty  will  even  then  be  experienced  on  account 
Swedish  n  to  wbich  it  v/ill  reach  if  the  tongue  of  the  variability  of  the  same  speaker  in  his 
be  slightlv  raised  into  the  A  position.  It  is  vowel  quality  ifor  differences  of  pitch  and 
necessarv'to  bear  these  facts  in  mind  when  expression,  the  want  of  habit  to  maintain  the 
following  the  text,  where  U  is  only  almost,  not  position  of  the  mouth  unmoved  for  a  sufficient 
ciuite  =  («/ inywo/-,  which  is  the  long  somid  of  (^  length  of  time  to  complete  an  observation 
in  ;/////,  and  is  duller  than  ou  in  poof  or  Frencli  satisfactorily,  and,  worst  of  all,  the  involuntary 
oti.  in  puu/c.—  Tniiis/i'Jor.i  tendency  cf  the  organs  to  accommodate  thein- 
t  [Great  difficulties  lie  in  the  way  of  carry-  selves  to  the  pitch  of  the  fork  presented.  Ccm- 
ing  out  this  recommendation.  The  car  of  pare  note  *  p.  105c. —  Translator^ 
philologists  and  even  of  those  who  are  readily 


CHAP.  V.   7.  VOWEL  QUALITIES  ()E  ToXE.  107 

Experiments    liereufter    described  show  that  the    resonaiur    of    tliis  siiii^h'   toiu-  is 
suthcient  to  characterise  the  vowels  above  mentioned. 

The  second  series  of  vowels  consists  of  A,  A,  E,  1.  The  lips  are  drawn  so  far 
apart  that  they  no  longer  contract  the  iss\iing  stream  of  air,  bnt  a  fresh  constric- 
tion is  formed  between  the  front  (middle)  parts  of  the  tongue  and  the  hard  jjalate, 
the  space  immediately  above  the  larynx  being  widened  by  depressing  the  root  of 
the  tongue,  and  hence  causing  the  larynx  to  rise  simultaneously.  The  form  of  the 
oral  cavity  consequently  resembles  a  bottle  with  a  narrow  neck.  The  belly  of  the 
bottle  is  behind,  in  the  pharynx,  and  its  neck  is  the  narrow  passage  between  the 
upper  surface  of  the  tongue  and  the  hard  palate.  In  the  above  series  of  letters, 
A,  E,  I,  these  changes  increase  until  for  I  the  internal  cavity  of  the  bottle  is  greatest 
and  the  neck  narrowest.  For  A  [the  broadest  French  e,  broader  than  e  in  there, 
and  nearly  as  broad  as  a  in  bat  lengthened,  with  which  the  name  of  their  city  is 
pronounced  by  the  natives  of  Bath],  tlie  whole  channel  is,  however,  tolerably  wide,  *fi 
so  that  it  is  quite  easy  to  see  down  to  the  larynx  when  the  laryngoscope  is  used. 
Indeed  this  vowel  gives  the  very  best  position  of  the  mouth  for  the  application  of 
this  instrument,  Ijecause  the  root  of  the  tongue,  which  impedes  tlie  view  when  A 
is  uttered,  is  depressed,  and  the  observer  can  see  over  and  past  it. 

When  a  bottle  with  a  long  narrow  neck  is  used  as  a  resonance  chamber,  two 
simple  tones  are  readily  discovered,  of  which  one  can  be  regarded  as  the  pi-oper 
tone  of  the  belly,  and  the  other  as  that  of  the  neck  of  the  bottle.  Of  course  the 
air  in  the  belly  cannot  vibrate  quite  independently  of  that  in  the  neck,  and  both 
proper  tones  in  question  must  consequently  be  difterent,  and  indeed  somewhat 
deeper  than  they  would  be  if  belly  and  neck  were  separate  and  had  their  resonance 
examined  independently.  The  neck  is  approximately  a  short  pipe  open  at  botli 
ends.  To  be  sure,  its  inner  end  debouches  into  the  cavity  of  the  bottle  instead  of 
the  open  air,  but  if  the  neck  is  very  narrow,  and  the  belly  of  the  bottle  very  wide, 
tlie  latter  may  be  looked  upon  in  some  respect  as  an  open  space  with  regard  to  the  H 
vibrations  of  the  air  inclosed  in  the  neck.  These  conditions  are  best  satisfied  for 
I,  in  which  the  length  of  the  channel  between  tongue  and  palate,  measured  from 
the  upper  teeth  to  the  back  edge  of  the  bony  palate,  is  aboiit  6  centimetres  [2 '36 
inches].  An  open  pipe  of  this  length  when  blown  would  give  e"",  while  the 
observations  made  for  determining  the  tone  of  loudest  resonance  for  I  gives  nearly 
(/"",  which  is  as  close  an  agreement  as  could  ])0ssibly  have  been  expected  in  such 
an  irregularly  shaped  pipe  as  that  formed  by  the  tongue  and  palate. 

In  accordance  with  these  experiments  the  vowels  A,  E,  I,  have  each  a  higher 
and  a  deeper  resonance  tone.  The  higher  tones  continue  the  ascending  series  of 
the  proper  tones  of  the  vowels  U,  0,  A.  By  means  of  tuning-forks  I  found  for  A 
a  tone  between  r/'"  and  a!"\},  and  for  E  the  tone  h"'\).  I  had  no  fork  suitable  for 
I,  })ut  l)y  means  of  the  whistling  noise  of  the  air,  to  be  considered  presently 
(p.  1086),  its  proper  tone  was  determined  with  tolerable  exactness  to  be  <}"" . 

The  deeper  proper  tones  which  are  due  to  the  back  part  of  the  oral  cavity  areH 
rather  more  difficult  to  discover.  Tuning-forks  may  be  used,  but  the  resonance  is 
comparatively  weak,  because  it  must  be  conducted  through  the  long  narrow  neck 
of  the  air  chamber.  It  must  further  be  remembered  that  this  resonance  only 
occurs  during  the  time  that  the  corresponding  vowel  is  gently  whispered,  and  dis- 
appears as  soon  as  the  whisper  ceases,  because  the  form  of  the  cliamber  on  which 
the  resonance  depends  then  immediately  changes.  The  tuning-forks  after  being 
struck  must  be  brought  as  close  as  possible  to  the  opening  of  the  air  chamber 
which  lies  behind  the  upper  teeth.  By  this  means  I  found  d"  for  A  and./''  for  E. 
For  I,  direct  observation  with  tuning-forks  was  not  possible  ;  but  from  the  upper 
partial  tones,  I  conclude  that  its  proper  tone  is  as  deep  as  that  of  V,  or  near  ./. 
Hence,  when  we  pass  from  A  to  I,  these  deeper  proper  tones  of  the  oral  cavity  sink, 
and  the  higher  ones  rise  in  pitch. "' 

*  [Mr.  Graham  Bell,  the  inventor  of  the  mentioned  (p.  105^/,  note),  was  hi  tlie  habit  of 
Telephone,  son  of  the  Mr.  Melville  Bell  already       bringing  out  this  fact  by  placing  his  moiitii  iu 


108  VOWEL  QUALITIES  OF  TONE.  part  i. 

F<3r  the  third  series  of  vowels  from  A  throug'li  0  [French  oi  in  jif-v,  ny  the 
deeper  e«  in  ^-'fH/vZe],  towards  U  [French  u  in  jm,  which  is  rather  deeper  than  the 
German  sound],  we  have  the  same  internal  positions  of  the  mouth  as  in  the  hist- 
named  series  of  vowels.  For  U  the  mouth  is  placed  in  nearly  the  same  position 
as  for  a  vowel  lying  between  E  and  I,  and  for  0  as  for  an  E  which  inclines  towards 
A.  In  addition  to  the  constriction  between  the  tongue  and  palate  as  in  the  second 
series,  we  have  also  a  constriction  of  the  lij^s,  which  are  made  into  a  sort  of  tube, 
forming  a  front  prolongation  of  that  made  by  the  tongue  and  palate.  The  air 
chamber  of  the  mouth,  therefore,  in  this  case  also  resembles  a  bottle  with  a  neck, 
but  the  neck  is  longer  than  for  the  second  series  of  vowels.  For  I  the  neck  was 
6  centimetres  (2-36  inches)  long,  for  LI,  measured  from  the  front  edge  of  the  ujjper 
teeth  to  the  commencement  of  the  soft  palate,  it  is  S  centimetres  {Z-\o  inches). 
The  pitch  of  the  higher  proper  tone  corresponding  to  the  resonance  of  the  neck 

Umust  be,  therefore,  about  a  Fourth  deeper  than  for  I.  If  both  ends  were  free,  a  pipe 
of  this  length  would  give  //",  according  to  the  usual  calciilation.  In  reality  it 
resounded  for  a  fork  lying  between  //'"  and  c?'"!?,  a  divergence  similar  to  that 
found  for  I,  and  also  probably  attributable  to  the  back  end  of  the  tube  debouching 
into  a  wider  but  not  quite  open  space.  The  resonance  of  the  back  space  has  to  be 
observed  in  the  same  way  as  for  the  I  series.  For  O  it  is/',  the  same  as  for  E, 
and  for  U  it  is  /,  the  same  as  for  I. 

The  fact  that  the  cavity  of  the  mouth  for  different  vowels  is  tuned  to  different 
pitches  was  first  discovered  by  Bonders,*  not  with  the  help  of  tuning-forks,  but  by 
the  whistling  noise  produced  in  the  mouth  by  Avhispering.  The  cavity  of  the 
mouth  thiis  reinforces  by  its  resonance  the  corresponding  tones  of  the  windrush, 
which  are  produced  partly  in  the  contracted  glottis.f  and  partly  in  the  forward 
contracted  passages  of  the  mouth.  In  this  way  it  is  not  usual  to  obtain  a  complete 
musical  tone  ;  this  only  happens,  without  sensible  change  of  the  vowel,  for  U  and 

51  U,  Avhen  a  real  whistle  is  produced.  This,  however,  would  be  a  fault  in  speaking. 
We  have  rather  only  such  a  degree  of  reinforcement  of  the  noise  of  the  air  as 
occiu's  in  an  organ  pipe,  which  does  not  speak  well,  either  from  a  badly-constructed 
lip  or  an  insufficient  pressure  of  wind.  A  noise  of  this  kind,  although  not  In-ought 
up  to  being  a  complete  musical  tone,  has  nevertheless  a  tolerably  determinate 
pitch,  which  can  be  estimated  by  a  practised  ear.  But,  as  in  all  cases  where  tones 
of  very  different  qualities  have  to  be  compared,  it  is  easy  to  make  a  mistake  in  the 
Octave.     However,  after  some  of  the  important  pitches  have  been  detemiined  by 

the  required  positions  and  then  tapping  agaiiij^t  Cbr.  Hellwag,  De  Fonai(tione  Loquelai-   Diss., 

a  finger  placed  just  in  front  of  tlie  upper  teeth,  Tubingae,     1710. — Florcke,      Keue      Berliner 

for  the  higher  resonance,  and  placed  against  Monatssclirift,  Sept.  1803,  Feb.   1804. — Olivier 

the  neck,  just  above  the  larynx,  for  the  lower.  Ortho-ejm-ijraphischcs     Elementar-lVcrk,    1804, 

He  obligingly  performed  the  experiment  several  part  iii.  p.  21. 

times  privately  before  me,  and  the  successive  +  In  whispering,  the  vocal  chords  are  kept 
alterations  and  differences  in  their  direction  close,  but  the  air  passes  through  a  small 
€T  were  striking.  The  tone  was  dull  and  like  triangular  opening  at  tbe  back  part  of  the 
a  wood  harmonica.  Considerable  dexterity  glottis  between  tbe  arytenoid  cartilages.  [Ac- 
seemed  necessary  to  produce  the  effect,  and  I  cording  to  Czermak  {Sitzunijsheriddr,  Wiener 
could  not  succeed  in  doing  so.  He  carried  out  Akad.,  Matb.-Naturw.  CI.  April  29,  1858, 
the  experiment  much  further  than  is  suggested  p.  576)  the  vocal  chords  as  seen  through  the 
in  tbe  text,  embracing  tbe  whole  nine  positions  lar3-ngos€ope  are  not  quite  close  for  whisper, 
of  tbe  tongue  in  bis  father's  vowel  scheme,  but  are  nicked  in  the  middle.  Merkel  {Die 
and  obtaining  a  double  resonance  in  each  case.  Funktionm  da  menschlichcn  Schhnul-  unci 
This  fact  is  stated,  and  the  various  vowel  Kehlkopfcs.  .  .  .  nach  eiyenen  2)haryy)f/o-  wnd 
theories  appreciated  in  j\Ir.  Graham  Bell's  larynrjoakopischen  Untersuchunge/h  Leipzig, 
paper  on  '  Vowel  Theories  '  read  before  the  1862,  p.  77)  distinguishes  two  kinds  of  wbisper- 
Americau  National  Academy  of  Arts  and  ing:  (1)  the  loud,  in  wbichtbeopening  between 
Sciences,  April  15,  1879,  and  printed  in  the  tbe  chords  is  from  ^  to  f  of  a  line  wide,  pro- 
Amcrican  Journal  of  Otology,  vol.  i.  July  ducing  no  resonant  vibrations,and  that  between 
1879. — Translatm:]  the  arytenoids  is  somewhat  wider:  (2)  tbe 
'*  Archiv  far  die  Holldndischen  Bcitrdge  gentle,  in  which  tbe  vowel  is  commenced  as  in 
fiir  Natur-und  Hcilkundc  von  Bonders  und  loud  speaking,  with  closed  glottis,  and,  after  it 
Berlin,  vol  i.  p.  157.  Older  incomplete  obser-  has  begun,  tbe  back  part  of  the  glottis  is 
vations  of  the  same  circumstance  in  Samuel  opened,  while  the  chords  remain  close  and 
Revher's     Mathcsis    Mosnicu.     Kiel.     1619. —  motionless.  —  Translator.} 


VOWEL  QUALITIES  OF  TONE. 


109 


tuuing-furks,  iuid  others,  us  V  and  ("),  by  allowing  the  whisper  tn  pass  into  a 
regnlar  whistle,  the  rest  are  easily  determined  by  arranging  tlieni  in  a  melodic 
progression  with  the  first.     Thns  the  series  :  — 


Clear    A                  i                        A 

K 

I                     1 

[ft  in  fath-er] 
d'" 

[a  in  mat] 

[e  in  there'] 

forms  an  ascending  minor  chord  of  fj  in  the  second  Inversion  f,  [with  the  Fifth  in 
the  bass,]  and  can  be  readily  compared  with  the  same  melodic  progression  on  the 
pianoforte.  I  was  able  to  determine  the  pitch  for  clear  A,  A,  and  E  by  tnning- 
forks,  and  hence  to  fix  that  for  I  also.* 

■"Tlie  statements  of  Bonders  differ  slightly  Bonders,  uot  having  been  assisted  by  tuning-  % 

from  mine,  partly  because  they  have  reference  forks,  was  not  always  able  to  determine  with 

to  Butch  x^ronunciation,  while  mine  refer  to  the  certainty  to  what  Octave  the  noises  he  heard 

North  German  vowels  ;    and  partly   because  should  be  assigned. 


Vowel 

Pitch  accord- 
ing to 
Bonders 

Pitch  accord- 

inj;  to 

Helmholtz 

/ 

U 
0 
A 

eV 

0 

ti 

E 

I 

'1% 

.(/"to,/"b 

["The  extreme  divergence  of  results  obtained 
by  different  investigators  shows  the  inherent 
difficulties  of  the  determination,  which  (as 
already  indicated)  arise  partly  from  different 
values  attributed  to  the  vowels,  partly  from  the 
difficulty  of  retaining  the  form  of  the  mouth 
steadily  for  a  sufficient  time,  partly  from  the 
wide  range  of  tones  which  the  same  cavity  of 
the  mouth  will  more  or  less  reinforce,  partly 
from  the  difficulty  of  judging  of  absolute  pitch 
in  general,  and  especially  of  the  absolute  pitch 
of  a  scarcely  musical  whisper,  and  other  causes. 
In  C.  L.  Merkel's  riiysiolocjic  dcr  mensch- 
Jkhen  Sprache  (Leipzig,  1866),  p.  47,  a  table  is 
given  of  the  results  of  Reyher,  Hellwag, 
Florcke,  and  Bonders  (the  latter  differing  ma- 
terially from  that  just  given  by  Prof.  Helm- 
holtz),  and  on  Merkel's  p.  109,  he  adds  his  last 


results.  These  are  reproduced  in  the  following 
table  with  the  notes,  and  their  pitch  to  the  ^ 
nearest  vibration,  taking  a'  440,  and  supposing 
equal  temperament.  To  these  I  add  the  re- 
sults of  Bonders,  as  just  given,  and  of  Helm- 
holtz,  both  with  pitches  similarly  assumed. 
Koenig  (Comptes  Jiendus,  April  25,  1870)  also 
gives  his  pitches  v/ith  exact  numbers,  reckoned 
as  Octaves  of  the  7th  harmonic  of  c'  256,  and 
hence  called  h\^,  although  they  are  nearer  the 
a  of  this  standard.  Reference  should  also  be 
made  to  Br.  Koenig's  paper  on  '  Manometric 
Flames'  translated  in  the  Philosophical  Maga- 
zine, 1873,  vol.  xlv.  pp.  1-18,  105-114.  Lastly, 
Br.  Moritz  Trautmann  [Anglia,  vol.  i.  p.  590) 
very  confidently  gives  results  utterly  different 
from  all  the  above,  which  I  subjoin  with  the 
pitch  as  before.     I  give  the  general  form  of 


Table 

OF  Vowel  Resonances. 

Observer. 

U 

O                  A 

A 

E 

I 

U 

o 

1.  Reyher    .     . 

fiai 

di  156        a  220 "\ 
*           1    c'262r 

rf|l56 

^349 

c"  523 

^#s? 

2.  Hellwag  .     . 

(■131 

4 139    !  /il85 
<?196     !  c'^262 

a  220 

b24:l     1    f'262 

b\,233 

3.  Florcke    .     . 

c  131 

,/  392 

«'440    :     «"523 

g'  392 

c'  330 

4.  Bonders  ac-^ 
cording      to  r 
Helmholtz  .J 

1 

r349 

d'29i       &'t,466 

!j"'  1568 

c"'i  1109  !  /""  1397 

a"  880 

r/196? 

+  d"587 

! 

5.  Bonders   ac-"^ 
cording      to  r 
Merkel    .     .J 

t'1651 

e 165         b 247 

c'  262    i  /"  698 

n'  440 

.^190 

/'175J 

: 

c"'till09 

6.  Helmholtz. 

U,/'175 

b'[,  466     b"'Q  932 

f/'"  1568 

b'"  1976 

d""  2349 

^"'1568 

Ou,  /'349 

-t-fr'587 

+  /349 

-f/175 

-f/175 

+/■'  349 

7.  Alerkel     .     . 

dl^l 

/iil85    ,A\rt220 

d"  587 

W,d"587 

a"  880 

a'  440 

0^,g  196 

A',  b  247 

or  a' 440 

E',e"659 

or  d'  294 

8.  Koenig,    7tli 

harmonics  . 

b\f22i 

b'[,  448 

b"h  896 

&"'k  1792  ]i""lj  3584 

0  ,fl"'1568 

9.  Trautmann . 

f'698 

O\c"'1047/"'1397 

^K? 

E\fl"'1760l  f '"  2794  ^*"' 1976 

0',«"880 

E',c""2093[ 

O',a"'1760 

no 


VOWEL  QUALITIES  OF  TONE. 


For  U  it  is  also  by  no  means  easy  to  find  the  pitch  of  the  resonance  by  a  fork, 
i\s  the  snialhiess  of  the  opening  makes  the  resonance  weak.  Another  phenomenon 
has  guided  me  in  this  case.  If  I  sing  the  scale  from  c  upwards,  uttering  the  vowel 
U  for  each  note,  and  taking  care  to  keep  the  quality  of  the  vowel  correct,  and  not 
allowing  it  to  pass  into  0,*  I  feel  the  agitation  of  the  air  in  the  mouth,  and  even 
on  the  drums  of  both  ears^  where  it  excites  a  tickling  sensation,  most  powerfully 
when  the  voice  reaches  /.  As  soon  as  ,/'  is  passed  the  quality  changes,  the  strong 
agitation  of  the  air  in  the  mouth  and  the  tickling  in  the  ears  cease.  For  the  note 
/  the  phenomenon  in  this  case  is  the  same  as  if  a  spherical  resonance  chamber 
were  placed  before  a  tongue  of  nearly  the  same  pitch  as  its  proper  tone.  In  this 
case  also  we  have  a  powerful  agitation  of  the  air  within  the  sphere  and  a  sudden 
alteration  of  quality  of  tone,  on  passing  from  a  deeper  pitch  of  the  mass  of  air 
through  that  of  the  tongue  to  a  higher.  The  resonance  of  the  mouth  for  U  is  thus 
«l  fixed  at ./'  with  more  certainty  than  by  means  of  tuning-forks.  But  we  often  meet 
with  a  U  of  higher  resonance,  more  resembling  0,  which  I  will  represent  by  the 
French  Ou.  Its  proper  tone  may  rise  as  high  as  f.\  The  resonance  of  the 
cavity  of  the  mouth  for  different  vowels  may  then  be  expressed  in  the  notes  as  follows: 


/ 


b'b  n 


d"" 

f 

0- 


Ou 


E 


0        U 


Tlie  mode  in  which  the  resonance  of  the  cavity  of  the  mouth  acts  upon  the 
quality  of  the  voice,  is  then  precisely  the  sanio  as  that  which  we  discovered  to 
exist  for  artificially  constructed  reed  pipes.  All  those  partial  tones  are  reinforced 
which  coincide  with  a  proper  tone  of  the  cavity  of  the  mouth,  or  have  a  pitch 
sufficiently  near  to  that  of  such  a  tone,  while  the  other  partial  tones  will  be  more 
or  less  damped.  The  damping  of  those  partial  tones  which  are  not  strengthened 
is  the  more  striking  the  narrower  the  opening  of  the  mouth,  either  between  the 
lips  as  for  U,  or  between  the  tongue  and  palate  as  for  I  and  U. 

These  differences  in  the  partial  tones  of  the  different  vowel  sounds  can  be  easily 
and  clearly  recognised  by  means  of  resonators,  at  least  within  the  once  and  twice 
accented  Octaves  [:264  to  1056  vib.].  For  example,  apply  a  b'\f  resonator  to  the 
ear,  and  get  a  bass  voice,  that  can  preserve  pitch  well  and  form  its  vowels  with 
purity,  to  sing  the  series  of  vowels  to  one  of  the  harmonic  under  tones  of  h'\),  such 
as  b\f,  e\),  B\f,  G\),  E\).  It  will  be  found  that  for  a  pure,  full-toned  0  the  y\)  of 
lithe  resonator  will  bray  violently  into  the  ear.  The  same  upper  partial  tone  is 
still  very  powerful  for  a  clear  A  and  a  tone  intermediate  between  A  and  O,  but  is 
weaker  for  A,  E,  O,  and  weakest  of  all  for  U  and  I.  It  will  also  be  found  that 
the  resonance  of  0  is  materially  weakened  if  it  is  taken  too  dull,  approaching  U, 

*  [That  is,  according  to  the  previous  direc- 
tions, to  keep  the  tongue  altogether  depressed, 
in  the  position  for  av:  in  gnaw,  which  is  not 
natural  for  an  Englishman,  so  that  for  English 
00  in  too  we  may  expect  the  result  to  be  ma- 
terially different. —  Translator.'] 

t  Prof.  Helmholtz  may  mean  the  Swedish 
0,  see  note*  p.  106rf.  The  following  words  im- 
mediately preceding  the  notes,  which  occur 
in  the  3rd  German  edition,  appear  to  have 
been  accidentally  omitted  in  the  4th.  They 
are,  however,  retained  as  they  seem  necessary. 
—  Translator. ] 


the  vowel  at  the  head  of  each  column,  and 
when  the  writer  distinguishes  different  forms 
I  add  them  immediately  before  the  resonance 
note.  Thus  we  liave  Helmholtz's  Ou  between 
U  and  O  ;  Merkel's  O'^  between  0  and  A,  his 
obscure  A\  E'  and  clear  A',  E' ;  Trautmann's 
O'  =  Italian  open  0,  and  (as  he  says)  English 
a  in  all  (which  is,  however,  slightly  different), 
0'  ordinary  o  in  Berliner  oltnc,  E'  Berlin 
Schnee,  E'  French  pere  (the  same  as  A  ?) ,  O' 
Berlin  schon,  French  pen,  ()'  French  Icur.  Of 
course  this  is  far  from  exhausting  the  list  of 
vowels  in  actual  use. — Translator.] 


CHAP.    V.     /  , 


VOWEL  QUALITIES  OF  TONE.  Ill 


or  too  open,  becoming  A ^.  But  if  tlie  ll'\)  resonator  be  used,  whicli  is  an  Octave 
liigher,  it  is  the  vowel  A  that  excites  the  strongest  sympathetic  resonance  ;  while  0, 
Avhich  was  so  powerful  with  the  h'\}  resonator,  now  produces  only  a  slight  effect. 

For  the  high  upper  partials  of  A,  E,  I,  no  resonators  can  be  made  which  are 
capable  of  sensibly  reinforcing  them.  We  are,  then,  driven  principally  to  observa- 
tions made  with  the  unassisted  ear.  It  has  cost  me  much  trouble  to  determine  these 
strengthened  jjartial  tones  in  the  vowels,  and  I  was  not  acquainted  with  them  when 
my  previous  accounts  were  published.*  They  are  best  observed  in  high  notes  of' 
women's  voices,  or  the  falsetto  of  men's  voices.  The  upper  partials  of  high  notes 
in  that  j)art  of  the  scale  are  not  so  nearly  of  the  same  pitch  as  those  of  deeper  notes, 
iind  hence  they  are  more  readily  distinguished.  On  //[j,  for  example,  women's 
voices  could  easily  bring  out  all  the  vowels,  with  a  full  quality  of  tone,  but  at 
higher  pitches  the  choice  is  more  limited.  When  h'\)  is^sung,  then,  the  Twelfth/'" 
is  heard  for  the  broad  A,  the  double  Octave  h"'\)  for  E,  the  high  Third  d""  for  I,  H 
all  clearly,  the  last  even  piercingly.     [See  table  on  p.  124,  note.]  f 

Further,  I  should  observe,  that  the  table  of  notes  given  on  the  preceding  page, 
relates  only  to  those  kinds  of  vowels  which  appear  to  me  to  have  the  most  cha- 
racteristic quality  of  tone,  but  that  in  addition  to  these,  all  intermediate  stages 
are  possible,  passing  insensibly  from  one  to  the  other,  and  are  actually  used  partly 
in  dialects,  partly  by  particular  individuals,  partly  in  peculiar  pitches  while  singing, 
or  to  give  a  more  decided  character  while  whispering. 

It  is  easy  to  recognise,  and  indeed  is  sufficiently  well  known,  that  the  vowels 
with  a  single  resonance  from  U  through  0  to  clear  A  can  be  altered  in  continuous 
succession.  But  I  wish  further  to  remark,  since  doubts  have  been  thrown  on  the 
deep  resonance  I  have  assigned  to  U,  that  when  I  apply  to  my  ear  a  resonator 
tuned  to/',  and,  singing  upon/ori?[?  as  the  fundamental  tone,  try  to  find  the 
vowel  resembling  U  which  has  the  strongest  resonance,  it  does  not  answer  to  a 
dull  L",  but  to  a  U  on  the  wav  to  Qi.X  ^| 

Then  again  transitions  gje  possible  between  the  vowels  of  the  A — 0 — U  series 
and  those  of  the  A — () — U  series,  as  well  as  between  the  last  named  and  those  of 
the  A — E — I  series.  I  can  begin  on  the  position  for  U,  and  gradually  transform 
the  cavity  of  the  mouth,  already  narrowed,  into  the  tube-like  forms  for  O  and  li , 
in  which  case  the  high  resonance  becomes  more  distinct  and  at  the  same  time 
higher,  the  narrower  the  tube  is  made.  If  we  make  this  transition  while  applying 
a  resonator  between  />'[>  and  h"\f  to  the  ear,  we  hear  the  loudness  of  the  tone 
increase  at  a  certain  stage  of  the  transition,  and  then  diminish  again.  The  higher 
the  resonator,  the  nearer  must  the  vowel  approach  to  0  or  U.  With  a  proper 
]josition  of  the  mouth  the  reinforced  tone  may  be  brought  up  to  a  whistle.  Also 
in  a  gentle  whisper,  where  the  rustle  of  the  air  in  the  larynx  is  kept  very  weak,  so 
that  with  vowels  having  a  naiTOw  opening  of  the  mouth  it  can  be  scarcely  heard,  a 
strong  fricative  noise  in  the  opening  of  the  mouth  is  often  required  to  make  the 
vowel  audible.  That  is  to  say,  we  then  make  the  vowels  more  like  their  related  ^ 
consonants,  English  W  and  German  J  [English  Y]. 

Generally  speaking  the  vowels  §  with  double  resonance  admit  of  numerous 
modifications,  because  any  high  pitch  of  one  of  the  resonances  may  combine  with 
any  low  pitch  of  the  other.  This  is  best  studied  by  applying  a  resonator  to  the 
ear  and  trying  to  find  the  corresponding  vowel  degi-ees  in  the  thi'ee  series  which 
reinforce  its  tone,  and  then  endeavouring  to  pass  from  one  of  these  to  the  other  in 
such  a  way  that  the  resonator  should  have  a  reinforced  tone  throughout. 

*  Gelehrtc  Anzeigcn   der  Bayerischen   Aka-  J  [An  U  sound  verging  towards  O  is  gene- 

demie  der  Wissenschaften,  June  18,  1859.  rally  conceived  to   be   duller  not  brighter,  by 

t  [The   passage    '  In    these    experiments '  English  writers,  but  here  U  is  taken  as  the 

to  '  too  deep  to  be   sensible,'  p.   166-7  of  the  dullest  vowel.     This  remark  is  made  merely 

1st   English   edition,  is   here   cancelled,  and  to  prevent  confusion  with  English  readers. — 

p.  111b,  '  Further,  I  should  observe,'  to  p.  116a,  Translator.^ 

'high  tones  of  A,  E,  I,'  inserted  in  its  place  §  [Misprinted  Consonanten  in  the  German, 

from  the  4th  German  edition. — Translator.]  — Translator.] 


112  VOWEL  QUALITIES  OE  TONE.  pakt  i. 

Thus  the  resonator  //f>  answers  to  0,  to  an  Aii  and  to  an  E  which  resembles  A, 
and  these  sounds  may  pass  continuously  one  into  the  other. 

The  resonator  /''  answers  to  the  transition  Ou — O — E.  The  resonator  d"  to 
Oa— Ao — A.  In  a  similar  manner  each  of  the  higher  tones  may  he  connected 
with  various  deeper  tones.  Thus  assuming  a  position  of  the  mouth  which  would 
give  e'"  for  whistling,  we  can,  without  changing  the  pitch  of  the  fricative  sound  in 
the  mouth,  whisper  a  vowel  inclining  to  (')  or  inclining  to  U,  by  allowing  the 
fricative  sound  in  the  larynx  to  have  a  higher  or  deeper  resonance  in  the  back  part 
of  the  mouth.* 

In  comparing  the  strength  of  the  upper  partials  of  different  vowels  by  means  of 
resonators,  it  is  further  to  be  remembered,  that  the  reinforcement  by  means  of  the 
resonance  of  the  mouth  affects  the  prime  tone  of  the  note  produced  by  the  voice, 
as  well  as  the  upper  partials.  And  as  it  is  especially  the  vibrations  of  the  prime, 
^  which  by  their  reaction  on  the  vocal  chords  retain  these  in  regular  vibratory  motion, 
the  voice  speaks  much  more  powerfully,  when  the  prime  itself  receives  such  a 
reinforcement.  This  is  especially  observable  in  those  parts  of  the  scale  which 
the  singer  reaches  with  difficulty.  It  may  also  be  noted  with  reed  pipes  having 
metal  tongues.  When  a  resonance  pipe  is  applied  to  them  tuned  to  the  tone  of  the 
tongue,  or  a  little  higher,  extraordinarily  powerful  and  rich  tones  are  produced,  by 
means  of  strong  pressure  but  little  wind,  and  the  tongue  oscillates  in  large  ex- 
cursions either  way.  The  pitch  of  a  metal  tongue  becomes  a  little  flatter  than 
before.  This  is  not  perceived  with  the  human  voice  because  the  singer  is  able  to 
regulate  the  tension  of  the  vocal  chords  accordingly.  Thus  I  find  distinctly  that 
i\th'\f,  the  extremity  of  my  falsetto  voice,  I  can  sing  powerfully  an  0,  an  A,  and  an 
A  on  the  way  tf)  O,  which  have  their  resonance  at  this  pitch,  whereas  U,  if  it  is 
not  made  to  come  very  near  0,  and  I,  are  dull  and  uncertain,  with  the  expenditure 
of  more  air  than  in  the  former  case.  Regard  must  be  had  to  this  circumstance  in 
^  experiments  on  the  strength  of  upper  partials,  because  those  of  a  vowel  which  speaks 
powerfully,  may  become  proportionally  too  powerful,  as  compared  with  those  of  a 
vowel  which  speaks  weakly.  Thus  I  have  found  that  the  high  tones  of  the  soprano 
voice  which  lie  in  the  reinforcing  region  of  the  vowel  A  at  the  upper  extremity  of 
the  doubly-accented  [or  one-foot]  Octave,  when  sung  to  the  vowel  A,  exhibit  their 
higher  Octave  more  strongly  than  is  the  case  for  the  vowels  E  and  I,  which  do  not 
speak  so  well  although  the  latter  have  their  strong  resonance  at  the  upper  end  of 
the  thrice-accented  [or  six-inch]  Octave. 

It  has  been  already  remarked  (p.  39c)  that  the  strength  and  amplitude  of 
sympathetic  vibration  is  aftected  by  the  mass  an'd  boundaries  of  the  body  which 
vibrates  sympathetically.  A  body  of  considerable  mass  which  can  perform  its 
vibrations  as  much  as  possible  without  any  hindrance  from  neighbouring  bodies, 
and  has  not  its  motion  damped  by  the  internal  friction  of  its  parts,  after  it  has 
once  been  excited,  can  continue  to  vibrate  for  a  long  time,  and  consequently,  if  it 
•fl  has  to  be  set  in  the  highest  degree  of  sympathetic  vibration,  the  oscillations  of  the 
exciting  tone  must,  for  a  comparatively  long  time,  coincide  with  those  proper 
vibrations  excited  in  itself.  That  is  to  say,  the  highest  degree  of  sympathetic 
resonance  can  be  produced  only  by  using  tones  which  lie  within  very  narrow  limits 
of  pitch.  This  is  the  case  with  tuning-forks  and  bells.  The  mass  of  air  in  the 
cavity  of  the  mouth,  on  the  other  hand,  has  but  slight  density  and  mass,  its  walls, 
so  far  as  they  are  composed  of  soft  parts,  are  not  capable  of  offering  much  resist 
ance,  are  imperfectly  elastic,  and  when  put  in  vibration  have  much  internal  friction 
to  stop  their  motion.  Moreover  the  vibrating  mass  of  air  in  the  cavity  of  the 
mouth  communicates  through  the  orifice  of  the  mouth  with  the  outer  air,  to  which 
it  rapidly  gives  off  large  parts   of  the  motion  it  has  received.     For  this  reason  a 

*  This  appears  to  me  to  meet  the  objec-  my  attention  to  the  habit  of  using  such  devia- 
tions which  were  made  by  Herr  G.  Engel,  in  tions  from  the  usual  quahties  of   vowels  in 
Reichart's  and  Du  Bois-Reymond's   Archiv.,  syllables  which  are  briefly  uttered. 
1869,  pp.  317-319.     Herr  J.  Stockhausen  drew 


CHAPV.  7.  MODIFICATIONS  OF  VOWEL  QUALITIES.  113 

vibratory  niotiou  once  excited  in  the  air  tilling  the  cavity  of  the  mouth  is  very 
rapidly  extinguished,  as  any  one  may  easily  observe  by  filliping  liis  cheek  with  a 
finger  when  the  mouth  is  put  into  different  vowel  positions.  We  thus  very  easily 
distinguish  the  pitch  of  the  resonance  for  the  various  transitional  degrees  from  O 
towards  V  in  one  direction  and  towards  A  in  the  other.  But  the  tone  dies  away 
rapidly.  The  various  resonances  of  the  cavity  of  the  mouth  can  also  be  made 
audible  by  rapping  the  teeth.  Just  for  this  reason  a  tone,  which  oscillates  approxi- 
mately in  agreement  with  the  few  vibrations  of  such  a  brief  resonance  tone,  will  be 
reinforced  by  sympathetic  vibration  to  an  extent  not  much  less  than  another  tone 
which  exactly  coincides  with  the  first ;  and  the  range  of  tones  which  can  thus 
be  sensibly  reinforced  by  a  given  position  of  the  mouth,  is  rather  considerable.^' 
This  is  confirmed  by  experiment.  When  I  apply  a  b'\f  resonator  to  the  right, 
and  ail /"  resonator  to  the  left  ear  and  sing  the  vowel  0  on  B\j,  I  find  a  reinforce- 
ment not  only  of  the  4th  partial  h'\f  which  answers  to  the  proper  tone  of  the  H 
cavity  of  the  mouth,  but  also,  very  perceptibly,  though  considerably  less,  of  /'. 
the  6th  partial,  also.  If  I  then  change  0  into  an  A,  until  /"  finds  its  strongest 
resonance,  the  reinforcement  of  b'\^  does  not  entirely  disappear  although  it  becomes 
much  less. 

The  position  of  the  mouth  from  0  to  0^  appears  to  be  that  which  is  most 
favourable  for  the  length  of  its  proper  tone  and  the  pi-oduction  of  a  resonance 
limited  to  a  very  naiTow  range  of  pitch.  At  least,  as  I  have  before  remarked,  for 
this  position  the  reinforcement  of  a  suitable  tuning-fork  is  most  powerful,  and  tap- 
ping the  cheek  or  the  lips  gives  the  most  distinct  tone.  If  then  for  0  the  rein- 
forcement by  resonance  extends  to  the  interval  of  a  Fifth,  the  intervals  will  be  still 
greater  for  the  other  vowels.  With  this  agree  experiments.  Apply  any  resonator 
to  the  ear,  take  a  suitable  under  tone,  sing  the  different  vowels  to  tliis  under  tone,  and 
let  one  vowel  melt  into  another.  The  greatest  reinforcements  by  resonance  take 
place  on  that  vowel  or  those  vowels,  for  which  one  of  the  characteristic  tones  in  51 
the  diagram  p.  100/y  coincides  with  the  proper  tone  of  the  resonator.  But  more  or 
less  considerable  reinforcement  is  also  observed  for  such  vowels  as  have  their  charac- 
teristic tones  at  moderate  differences  of  pitch  from  the  proper  tone  of  the  resonator, 
and  the  reinforcement  will  be  less  the  greater  these  differences  of  pitch. 

By  this  means  it  becomes  possible  in  general  to  distinguish  the  vowels  from 
each  other  even  when  the  note  to  which  they  are  sung  is  not  precisely  one  of  the 
harmonic  under  tones  of  the  vowels.  From  the  second  partial  tone  onwards,  the 
intervals  are  narrow  enough  for  one  or  two  of  the  partials  to  be  distinctly  reinforced 
by  the  resonance  of  the  mouth.  It  is  only  when  the  proper  tone  of  the  cavity  of 
the  mouth  falls  midw  ay  between  the  prime  tone  of  the  note  sung  by  the  voice  and 
its  higher  Octave,  or  is  more  than  a  Fifth  deeper  than  that  prime  tone,  that  the 
characteristic  resonance  will  be  weak. 

Now  in  speaking,  both  sexes  choose  one  of  the  deepest  positions  of  their  voice. 
Men  generally  choose  the  upper  half  of  the  great  (or  eight-foot)  Octave ;  and  H 
women  the  upper  half  of  the  small  (or  four-foot)  Octave,  t  With  the  exception  of 
U,  which  admits  of  fluctuations  in  its  proper  tone  of  nearly  an  Octave,  all  these 
pitches  of  the  speaking  voice  have  the  corresponding  proper  tones  of  the  cavity 
of  the  mouth  situated  within  sufficiently  narrow  intervals  from  the  upper  partials  of 
the  speaking  tone  to  create  sensible  resonance  of  one  or  more  of  these  partials, 
and  thus  characterise  the  vowel.  |  To  this  must  be  added  that  the  speaking  voice, 
probably  through  great  pressure  of  the  vocal  ligaments  upon  one  another,  converting 

*  On  this  subject  see   Appendix   X.,  and  of  certain  of  its  partials  with  exact  pitches 

the  corresponding  investigation  in  the  text  in  but  in   their   coming   near  enough   to   those 

Pai't  I.  Chap.  VI.  therein  referred  to.  pitches  to  receive  reinforcement,  and  that  the 

+  [That    is    both  use   their    '  lower  thick  '  character  of  a  vowel  quality  of  tone,  like  that 

register,  as  described  in  the  note  p.  101c?,  but  of  all  qualities  of  tone,  depends  not   on    the 

are  an  Octave  apart. —  Translator.]  absolute  pitch,  but  on  the  relative  force  of  the 

+  [Observe   here   that   the   quality   of  the  upper  partials.     As  Prof.   Helmholtz's  theory 

vowel  tone  is  not  made  to  consist  in  the  identity  has   often  been   grievously  misunderstood,   I 

I 


114  MODIFICATIONS  OF  VOWEL  QUALITIES.  part  i. 

them  into  striking  reeds,  has  a  jarring  quahty  of  tone,  that  is,  possesses  stronger 
upper  partials  than  the  singing  voice. 

In  singing,  on  the  other  hand,  especially  at  higher  pitches,  conditions  are  less 
favourable  for  the  characterisation  of  vowels.  Every  one  knows  that  it  is  generally 
much  more  difficult  to  understand  words  when  sung  than  when  spoken,  and  that 
the  difficulty  is  less  with  male  than  with  female  voices,  each  having  been  equally  well 
cultivated.  Were  it  otherwise,  '  books  of  the  words '  at  operas  and  concerts  would 
be  imnecessary.  Above  /',  the  characterisation  of  U  becomes  imperfect  even  if  it 
is  closely  assimilated  to  0.  But  so  long  as  it  remains  the  only  vowel  of  indetermi- 
nate sound,  and  the  remainder  allow  of  sensible  reinforcement  of  their  upper  partials 
in  certain  regions,  this  negative  character  will  distinguish  U.  On  the  other  hand 
a  soprano  voice  in  the  neighbourhood  of  /"  should  not  be  able  to  clearly  distinguish 
U,  0,  and  A  ;  and  this  agrees  with  my  own  experience.     On  singing  the  three  vowels 

U  in  inmiediate  succession,  the  resonance  ./'"'  for  A  will,  however,  be  still  somewhat 
clearer  in  the  cavity  of  the  mouth  when  tuned  for  ^"[>,  than  when  it  is  tuned  to  h'\) 
for  0.  The  soprano  voice  will  in  this  case  be  able  to  make  the  A  clearer,  by  eleva- 
ting the  pitch  of  the  cavity  of  the  mouth  towards  d'"  and  thus  making  it  approach 
to/'".  The  0,  on  the  other  hand,  can  be  separated  from  U  by  approaching  0,„  and 
giving  the  prime  more  decisive  force.  Nevertheless  these  vowels,  if  not  sung  in 
immediate  succession,  will  not  be  very  clearly  distinguished  by  a  listener  who  is 
unacquainted  with  the  mode  of  pronouncing  the  vowels  that  the  soprano  singer 
uses.* 

A  further  means  of  helping  to  discriminate  vowels,  moreover,  is  found  in  com- 
mencing them  powerfully.  This  depends  upon  a  general  relation  in  bodies  excited 
to  sympathetic  vibration.  Thus,  if  we  excite  sympathetic  vibration  in  a  suitable 
body  with  a  tone  somewhat  different  from  its  proper  tone,  by  commencing  it  suddenly 
with  great  power,  we  hear  at  first,  in  addition  to  the  exciting  tone  which  is  rein- 

^  forced  by  resonance,  the  proper  tone  of  the  sympathetically  vibrating  body.f  But 
the  latter  soon  dies  away,  while  the  first  i-emains.  In  the  case  of  tuning-forks  with 
large  resonator,  we  can  even  hear  beats  between  the  cwo  tones.  Apply  a  b'\j  resonator 
to  the  ear,  and  commence  singing  the  vowel  0  powerfully  on  g,  of  which  the  upper 
partials  g  and  d"  have  only  a  weak  lasting  resonance  in  the  cavity  of  the  mouth, 
and  you  may  hear  immediately  at  the  commencement  of  the  vowel,  a  short  shai-p 
beat  between  the  b'\)  of  the  cavity  of  the  mouth  and  of  the  resonator.  On  selecting 
another  vowel,  this  l>'\)  vanishes,  which  shows  that  the  pitch  of  the  cavity  of  the 
mouth  helps  to  generate  it.  In  this  case  then  also  the  sudden  commencement  of 
the  tones  g'  and  d"  belonging  to  the  compound  tone  of  the  voice,  excites  the  inter- 
mediate proper  tone  li'\f  of  the  cavity  of  the  mouth,  which  rapidly  fades.  The 
same  thing  may  be  observed  for  other  pitches  of  the  resonator  used,  when  we  sing 
notes,  powerfully  commenced,  which  have  upper  partials  that  are  not  reinforced  by 
the  resonator,  provided  that  a  vowel  is  chosen  wuth  a  characteristic  pitch  which 

\\  answers  to  the  pitch  of  the   resonator.     Hence  it  results  that  when  any  vowel  in 

,  any  pitch  is  powerfully  commenced,  its  characteristic  tone  becomes  audible  as  a 
short  beat.  By  this  means  the  vowel  may  be  distinctly  characterised  at  the 
moment  of  commencement,  even  when  it  becomes  intermediate  on  long  con- 
tinuance. But  for  this  purpose,  as  already  remarked,  an  exact  and  energetic  com- 
mencement is  necessary.  How  much  such  a  commencement  assists  in  rendering 
the  words  of  a  singer  intelligible  is  well  known.  For  this  reason  also  the  vocal- 
isation of  the  briefly-uttered  words  of  a  reciting  parlando,  is  more  distinct  than 
that  of  sustained  song.  J 

draw  particular  attention  to  the  point  in  this  may  make  in  the  vowels  in  English,  German, 

place.     See  also  the  table  which  I  have  added  French  and  Italian,  at  different  pitches,  so  as 

in  a  footnote  on  p.  124(1.— Translalor.]  to  remain  intelligible.— r/vow^rtiJor.] 

*  [In  my  Fronuncicition  for  Singers  (Cur-  f  See  the  mathematical  statement  of  this  pro- 
wen,  1877),  and  my  Speech  in  Song  (Novello,  cess  in  App.  IX.,  remarks  on  equations  4  to  46. 
1878)  I  have  endeavoured  to  give  a  popular  \  The  facts  here  adduced  meet,  I  think,  the 
explanation  of  the  alterations  which  a  singer  objections  brought  against  my  vowel  theory  by 


CHAP.  V,  7.  CHARACTERISTICS  OF  VOWELS.  115 

Moreover  vowels  admit  of  other  kinds  of  alterations  in  their  ([nalities  of  tone, 
conditioned  by  alterations  of  their  characteristic  tones  within  certain  limits.  Thus 
the  resonating  capability  of  the  cavity  of  the  mouth  may  undergo  in  general  altera- 
tions in  strength  and  definition,  which  woiild  render  the  character  of  the  various 
vowels  and  their  diflference  from  one  another  in  general  more  or  less  conspicuous 
or  obscure.  Flaccid  soft  walls  in  any  passage  with  sonorous  masses  of  air,  are 
generally  prejudicial  to  the  force  of  the  vibrations.  Partly  too  much  of  the  motion 
is  given  off'  to  the  outside  through  the  soft  masses,  partly  too  much  is  destroj^ed  by 
friction  within  them.  Wooden  organ  pipes  have  a  less  energetic  quality  of  tone 
than  metal  ones,  and  those  of  pasteboard  a  still  duller  quality,  even  when  the 
mouthpiece  remains  unaltered.  The  walls  of  the  human  throat,  and  the  cheeks, 
are,  however,  much  more  yielding  than  pasteboard.  Hence  if  the  tone  of  the  voice 
with  all  its  partials  is  to  meet  with  a  powerful  resonance  and  come  out  unweakened, 
these  most  flaccid  parts  of  the  passage  for  our  voice,  must  be  as  much  as  possible  H 
thrown  out  of  action,  or  else  rendered  elastic  by  tension,  and  in  addition  the  passage 
must  be  made  as  short  and  wide  as  possible.  The  last  is  effected  by  i-aising  the 
larynx.  The  soft  wall  of  the  cheeks  can  be  almost  entirely  avoided,  by  taking  care 
that  the  rows  of  teeth  are  not  too  far  apart.  The  lips,  when  their  co-operation  is 
not  necessary,  as  it  is  for  0  and  tJ,  may  be  held  so  far  apart  that  the  sharp  firm 
edges  of  the  teeth  define  the  orifice  of  the  mouth.  For  A  the  angles  of  the  mouth 
can  be  drawn  entirely  aside.  For  0  they  can  be  firmly  stretched  by  the  muscles 
above  and  below  them  {levator  angtdi  oris  and  triangularis  menti),  which  then  feel 
like  stretched  cords  to  the  touch,  and  can  be  thus  pressed  against  the  teeth,  so  that 
this  part  of  the  margin  of  the  orifice  of  the  mouth  is  also  made  sharp  and  capable 
of  resisting. 

In  the  attempt  to  produce  a  clear  energetic  tone  of  the  voice  we  also  become 
aw^are  of  the  tension  of  a  large  number  of  muscles  lying  in  front  of  the  throat, 
both  those  which  lie  between  the  under  jaw  and  the  tongue-bone  and  help  to  form  ^ 
the  floor  of  the  cavity  of  the  mouth  {rnylohyoideus,  geniohyoideus,  and  perhaps 
also  hiventer),  and  likewise  those  which  run  down  near  the  larynx  and  air  tubes,  and 
draw  down  the  tongue-bone  {sternohyoidetis,  stemothyroideus  and  thyrohyoideus). 
Without  the  counteraction  of  the  latter,  indeed,  considerable  tension  of  the  former 
would  be  impossible.  Besides  this  a  contraction  of  the  skin  on  both  sides  of  the 
larynx  which  takes  place  at  the  commencement  of  the  tone  of  the  voice,  shows  that 
the  omohyoidens  muscle,  which  runs  obliquely  down  from  the  tongue-bone  back- 
wards to  the  shoulder-blade,  is  also  stretched.  Without  its  co-operation  the  muscles 
arising  from  the  under  jaw  and  breast-bone  would  draw  the  larynx  too  far  forwards. 
Now  the  greater  part  of  these  muscles  do  not  go  to  the  larynx  at  all,  but  only  to 
the  tongue-bone,  from  which  the  larynx  is  suspended.  Hence  they  cannot  directly 
assist  in  the  formation  of  the  voice,  so  far  as  this  depends  upon  the  action  of  the 
larynx.  The  action  of  these  muscles,  so  far  as  I  have  been  able  to  observe  it  on 
myself,  is  also  much  less  when  I  utter  a  dull  guttural  A,  than  when  I  endeavour  to  ^\ 
change  it  into  a  ringing,  keen  and  powerfully  penetrating  A.  Ringing  and  keen, 
applied  to  a  quality  of  tone,  imply  many  and  powerful  upper  partials,  and  the 
stronger  they  are,  of  course  the  more  marked  are  the  difterences  of  the  vowels 
which  their  own  differences  condition.  A  singer,  or  a  dcclaimer,  will  occasionally 
interpose  among  his  bright  and  rich  tones  others  of  a  duller  character  as  a  contrast. 
Sharp  characterisation  of  vowel  quality  is  suitable  for  energetic,  joyful  or  vigorous 
frames  of  mind  ;  indifterent  and  obscure  quality  of  tone  for  sad  and  troubled,  or  taci- 
turn states.  In  the  latter  case  speakers  like  to  change  the  proper  tone  of  the  vowels, 
by  drawing  the  extremes  closer  to  a  middle  Ab  (say  the  short  German  E  [the  final 

Herr  E.  v.  Quanten  (PoggendorfE's  AnnciL,  article,  pp.  724-741,  with  especial  reference  to 
vol.  cliv.  pp.  272  and  522),  so  far  as  they  do  not  it.  In  consequence  of  the  new  matter  added 
rest  upon  misconceptions.  [In  the  1st  edition  by  Prof.  Helmlioltz  in  his  4th  German  edition 
of  this  translation,  during  the  printing  of  which  here  followed,  this  article  is  omitted  from  the 
V.  Quanten's  first  paper  appeared,  I  added  an       present  edition. — Translator.] 

I  2 


116  VOWEL  QUALITIES  OF  TONE.  part  i. 

English  obscure  A  in  uJea]),  and  hence  select  somewhat  deeper  tones  in  place  of  the 
high  tones  of  A,  E,  I. 

A  peculiar  circumstance  must  also  be  mentioned  which  distinguishes  the 
human  voice  from  all  other  instruments  and  has  a  peculiar  relation  to  the  human 
ear.  Above  the  higher  reinforced  partial  tones  of  I,  in  the  neighbourhood  of  e"" 
uptoy"[26-iO  to  3168  vib.]  the  notes  of  a  pianoforte  have  a  peculiar  cutting 
effect,  and  we  might  be  easil}  led  to  believe  that  the  hammers  were  too  hard,  or 
that  their  mechanism  somewhat  differed  from  that  of  the  adjacent  notes.  But  the 
phenomenon  is  the  same  on  all  pianofortes,  and  if  a  very  small  glass  tube  or  sphere 
is  applied  to  the  ear,  the  cutting  effect  ceases,  and  these  notes  become  as  soft  and  weak 
as  the  rest,  but  another  and  deeper  series  of  notes  now  becomes  stronger  and  more 
cutting.  Hence  it  follows  that  the  human  ear  by  its  own  resonance  favours  the  tones 
between  e""  and  c/"",  or,   in  other  words,  that  it  is  tuned  to  one  of  these  pitches.* 

^[  These  notes  produce  a  feeling  of  pain  in  sensitive  ears.  Hence  the  upper  partial 
tones  which  have  nearly  this  pitch,  if  any  such  exist,  are  extremely  prominent 
and  affect  the  ear  powerfully.  This  is  generally  the  case  for  the  human  voice  when 
it  is  strained,  and  will  help  to  give  it  a  screaming  effect.  In  powerful  male  voices 
singing  forte,  these  partial  tones  sound  like  a  clear  tinkling  of  little  bells,  accom- 
panying the  voice,  and  are  most  audible  in  choruses,  when  the  singers  shout  a 
little.  Every  individual  male  voice  at  such  pitches  produces  dissonant  upper  partials. 
When  basses  sing  their  high  /,  the  7th  partial  tone  f  is  d"",  the  8th  e"" ,  the 
9th /""J,  and  the  10th  f"j^.  Now,  if  e""  and  /""|  are  loud,  and  J""  and  /"jf, 
though  weaker,  are  audible,  there  is  of  course  a  sharp  dissonance.  If  many  voices 
are  sounding  together,  producing  these  upper  partials  with  small  differences  of 
pitch,  the  result  is  a  very  peculiar  kind  of  tinkling,  which  is  readily  recognised  a 
second  time  when  attention  has  been  once  drawn  to  it.  I  have  not  noticed  any 
difference  of  effect  for  different  vowels  in  this  case,  but  the  tinkling  ceases  as  soon 

ej  as  the  voices  are  taken  ^jmwio  ;  although  the  tone  produced  by  a  chorus  will  of 
course  still  have  considerable  power.  This  kind  of  tinkling  is  peculiar  to  human 
voices ;  orchestral  instruments  do  not  produce  it  iu  the  same  way  either  so  sensibly 
or  so  powerfully.  I  have  never  heard  it  from  any  other  musical  instrument  so 
clearly  as  from  human  voices. 

The  same  upper  partials  are  heard  also  in  soprano  voices  when  they  sing  forte  ; 
in  harsh,  uncertain  voices  they  are  tremulous,  and  hence  show  some  resemblance 
to  the  tinkling  heard  in  the  notes  of  male  voices.  But  I  have  heard  them  brought 
out  with  exact  purity,  and  continue  to  sovind  on  perfectly  and  (luietly,  in  some 
steady  and  harmonious  female  voices,  and  also  in  some  excellent  tenor  voices.  In 
the  melodic  progression  of  a  voice  part,  I  then  hear  these  high  upper  partials  of 
the  four-times  accented  Octave,  falling  and  rising  at  different  times  within  the 
compass  of  a  minor  Third,  according  as  different  upper  partials  of  the  notes  sung 
enter  the  region  for  which  our  ear  is  so  sensitive.     It  is  certainly  remarkable  that 

5j  it  should  be  precisely  the  human  voice  which  is  so  rich  in  those  upper  partials  for 
which  the  human  ear  is  so  sensitive.  Madame  E.  Seller,  however,  remarks  that 
dogs  are  also  very  sensitive  for  the  high  e""  of  the  violin. 

This  reinforcement  of  the  upper  partial  tones  belonging  to  the  middle  of  the 
four-times  accented  Octave,  has,  however,  nothing  to  do  with  the  characterisation 
of  vowels.  I  have  mentioned  it  here,  merely  because  these  high  tones  are  readily 
remarked  in  investigations  into  the  vowel  qualities  of  tone,  and  the  observer  must 
not  be  misled  to  consider  them  as  peculiar  characteristics  of  individual  vowels. 
They  are  simply  a  characteristic  of  strained  voices. 

The  humming  tone  heard  when  singing  with  closed  mouth,  lies  nearest  to  U. 

*  I  have  lately  found  that  my  right  ear  is  merely  applying  a  short  paper  tube  to  the  en- 
most  sensitive  for  /"",  and  my  left  for  c"".  trance  of  my  ear,  this  chirp  is  rendered  extra- 
When  I  drive  air  into  the  passage  leading  to  the  ordinarily  weak.  ^  ^ 
tympanum,  the  resonance  descends  to  0'"%  and  f  [The  first  six  partial  tones  are  e  ,  c  ,  ^^  , 
g"'i.  The  chirp  of  the  cricket  corresponds  e'",  g"%,  b'",  the  seventh  is  27  cents  flatter 
precisely  to   the    higher    resonance,   and   on  than  d"". — Translator.~\ 


CHAP.  V.   7. 


VOWEL  QUALITIES  OE  l^ONE. 


117 


This  hum  is  used  in  uttering  the  consonants  M,  N  and  N*-'.  The  size  of  the  exit 
of  the  air  (the  nostrils)  is  in  this  case  much  smaller  in  comparison  with  the 
resonant  chamber  (the  internal  nasal  cavity)  than  the  opening  of  the  lips  for  U  in 
comparison  with  the  corresponding  resonant  chamber  in  the  mouth.  Hence,  in 
humming,  the  peculiarities  of  the  U  tone  are  much  enhanced.  Thus  although 
upper  partials  are  present,  even  iip  to  a  considerably  high  pitch,  yet  they  decrease 
in  strength  as  they  rise  in  pitch  much  faster  than  for  U.  The  upper  Octave  is 
tolerably  strong  in  humming,  but  all  the  higher  partial  tones  are  weak.  Humming 
in  the  N-position  differs  a  little  from  that  in  the  M-position,  by  having  its  upper 
partials  less  damped  than  for  M.  But  it  is  only  at  the  instant  when  the  cavity  of 
the  mouth  is  opened  or  closed  that  a  clear  difference  exists  between  these  conso- 
nants. We  cannot  enter  into  the  details  of  the  com]wsition  of  the  sound  of  the 
other  consonants,  because  they  produce  noises  which  have  no  constant  pitch,  and 
are  not  musical  tones,  to  which  we  have  here  to  confine  our  attention.  *fl 

The  theory  of  vowel  sounds  here  explained  may  be  confirmed  by  experiments 
with  artificial  reed  pipes,  to  which  proper  resonant  chambers  are  attached.  This 
was  first  done  by  Willis,  who  attached  reed  pipes  to  cylindrical  chambers  of  variable 
length,  and  produced  different  tones  by  increasing  the  length  of  the  resonant  tube. 
The  shortest  tubes  gave  him  I,  and  then  E,  A,  0,  up  to  U,  until  the  tube  exceeded 
the  length  of  a  quarter  of  a  wave.  On  further  increasing  the  length  the  vowels 
returned  in  converse  order.  His  determination  of  the  pitch  of  the  resonant  pipes 
agrees  well  with  mine  for  the  deeper  vowels.  The  pitch  found  by  Willis  for  the 
higher  vowels  was  relatively  too  high,  because  in  this  case  the  length  of  the  wave 
was  smaller  than  the  diameter  of  the  tubes,  and  consequently  the  usual  calcula- 
tion of  pitch  from  the  length  of  the  tubes  alone  was  no  longer  applicable.  The 
vowels  E  and  I  were  also  far  from  accurately  resembling  those  of  the  voice,  because 
the  second  resonance  was  absent,  and  hence,  as  Willis  himself  states,  they  could 
not  be  w^ell  distinguished.*  ^' 


Pitch 

Pitch, 

Length  of  Tube 

\  owel 

In  the  Word 

Willis 

Helmholtz 

in  Inches 

0 

No 

c" 

,., 

[             4-7 

Ao 

Nought 

c"b 

c"b 

1            3-8 

Paw 

</" 

</' 

3-05 

A 

Part 

d"'\, 

d"'b 

2-2 

Pad 

/'" 

1-8 

E 

Pay 

d"" 

h"'\} 

!                 1-0 

Pet 

c'"" 

c"" 

0-6 

I 

See 

</"" 

0-38  (?) 

The  vowels  are  obtained  much  more  clearly  and  distinctly  with  properly  tuned 
resonators,  than  with  cylindrical  resonance  chambers.  On  applying  to  a  reed  pipe 
which  gave  f>\),  a  glass  resonator  tuned  to  b\),  I  obtained  the  vowel  U;  changing  H 
the  resonator  to  one  tuned  for  fy\),  I  obtained  0;  the  i"b  resonator  gave  a  rather 
close  A,  and  the  d'"  resonator  a  clear  A.  Hence  by  tuning  the  applied  chambers 
in  the  same  way  we  obtain  the  same  vowels  quite  independently  of  the  form  of  the 
chamber  and  nature  of  its  walls.     I  also  succeeded  in  |)roducing  various  grades  of 


*  [Probably  the  first  treatise  on  phonology 
in  which  Willis's  experiments  were  given  at 
length,  and  the  above  table  cited,  with  Wheat- 
stone's  article  from  the  London  and  Westmbi- 
stcr  Reviev\  which  was  kindly  brought  under 
my  notice  by  Sir  Charles  Wheatstone  himself, 
was  my  Alphabet  of  Nature,  London,  1845.  The 
table  includes  U  exemplified  by  but,  hoot,  with 
an  indefinite  length  of  pipe.  The  word  pad  is 
misprinted  paa.  in  all  the  German  editions  of 
Helmholtz  (even  the  4tb,  which  appeared  after 
the  correction  in  my  translation),  and  as  he 


therefore  could  not  separate  its  A  from  that  in 
ptdi,  he  gives  no  pitch.  It  is  really  the  nearest 
English  representative  of  the  German.  The 
sounds  in  noiujlit,  pan-,  which  Sir  John  Her- 
schel,  when  citing  Willis  (Art.  '  Sound,'  in 
Encijc.  MeiropoL,  par.  375),  could  not  distin- 
guish, were  probably  meant  for  the  broad 
Italian  open  O,  or  English  o  in  'more,  and  the 
English  aw  in  maio  respectively.  The  length 
of  the  pipe  in  inches  is  here  added  from  WiUis's 
paper.  I  have  heard  Willis's  experiments 
repeated  by  Whentatone.— Translator.] 


118  VOWEL  QUALITIES  OF  TONE.  parti. 

A,  0,  E,  and  1  with  the  same  reed  pipe,  by  applying  glass  spheres  into  whose  external 
opening  glass  tubes  were  inserted  from  6  to  10  centimetres  (2-36  to  3-94  inches)  in 
length,  in  order  to  imitate  the  double  resonance  of  the  oral  cavity  for  these 
vowels. 

Willis  has  also  given  another  interesting  method  for  producing  vowels.  If  a 
toothed  wheel,  with  many  teeth,  revolve  rapidly,  and  a  spring  be  applied  to  its 
teeth,  the  spring  will  be  raised  by  each  tooth  as  it  passes,  and  a  tone  will  be  pro- 
duced having  its  pitch  number  equal  to  the  number  of  teeth  by  which  it  has  been 
struck  in  a  second.  Now  if  one  end  of  the  spring  is  well  fastened,  and  the  spring 
be  set  in  vibi-ation,  it  will  its-elf  produce  a  tone  which  will  increase  in  pitch  as  the 
spring  diminishes  in  length.  If  then  we  turn  the  wheel  with  a  constant  velocity, 
and  allow  a  watch  spring  of  variable  length  to  strike  against  its  teeth,  we  shall 
obtain  for  a  long  spring  a  quality  of  tone  resembling  U,  and  as  we  shorten  the 

H  spring  other  qualities  in  succession  like  0,  A,  E,  I,  the  tone  of  the  spring  here 
playing  the  part  of  the  reinforced  tone  which  determines  the  vowel.  But  this 
imitation  of  the  vowels  is  certainly  much  less  complete  than  that  obtained  by  reed 
pipes.  The  reason  of  this  process  also  evidently  depends  upon  our  producing 
compound  tones  in  which  certain  upper  partials  (which  in  this  case  correspond  with 
the  proper  tones  of  the  spring  itself)  are  more  reinforced  than  others. 

Willis  himself  advanced  a  theory  concerning  the  nature  of  vowel  tones  which 
differs  from  that  I  have  laid  down  in  agreement  with  the  whole  connection  of  all 
other  acoustical  phenomena.  Willis  imagines  that  the  pulses  of  air  which  produce 
the  vowel  qualities,  are  themselves  tones  which  rapidly  die  away,  corresponding  to 
the  proper  tone  of  the  spring  in  his  last  experiment,  or  the  short  echo  jjroduced  by 
a  pulse  or  a  little  explosion  of  air  in  the  mouth,  or  in  the  resonance  chamber  of  a 
reed  pipe.  In  fact  something  like  the  sound  of  a  vowel  will  be  hcai-d  if  we  only 
tap  against  the  teeth  with  a  little  rod,  and  set  the  cavity  of  the  mouth  in  the  posi- 

11  tion  required  for  the  different  vowels.  Willis's  description  of  the  motion  of  sound 
for  vowels  is  certainly  not  a  great  way  from  the  truth  ;  but  it  only  assigns  the 
mode  in  which  the  motion  of  the  air  ensues,  and  not  the  corresponding  reaction 
which  this  produces  in  the  ear.  That  this  kind  of  motion  as  well  as  all  othei's 
is  actually  resolved  by  the  ear  into  a  series  of  partial  tones,  according  to  the  laws 
of  sympathetic  resonance,  is  shown  by  the  agreement  of  the  analysis  of  vowel 
qualities  of  tone  made  by  the  unarmed  ear  and  by  the  resonators.  This  will 
appear  still  more  clearh^  in  the  next  chapter,  where  experiments  will  be  described 
showing  the  direct  composition  of  vowel  qualities  from  their  partial  tones. 

Vowel  qualities  of  tone  consequently  are  essentially  distinguished  from  the 
tones  of  most  other  musical  instruments  b}'  the  fact  that  the  loudness  of  their 
partial  tones  does  not  depend  solely  upon  their  numerical  order  but  preponder- 
antly upon  the  absolute  pitch  of  those  partials.  Thus  when  I  sing  the  vowel  A  to 
the  note  ^!7,*  the  reinforced   tone  h"\y  is  the   12th  partial  of  the  compound  tone  ; 

Hand  when  I  sing  the  same  vowel  A  to  the  note  b'\),  the  reinforced  tone  is  still  h"\f^ 
but  is  now  the  2nd  partial  of  the  compound  tone  sung.t 


From  the  examples  adduced  to  show  the  dependence  of  quality  of  tone  from 
the  mode  in  which  a  musical  tone  is  conqjounded,  we  may  deduce  the  following 
general  rules  : — 

1.  Simple  Tones,  like  those  of  tuning-forks  applied  to  resonance  chambers  and 
wide  stopped  organ  pipes,  have  a  very  soft,  pleasant  sound,  free  from  all  i-oughness, 
but  wanting  in  power,  and  dull  at  low  pitches. 

2.  MusimJ  Tones,  which   are   accompanied  by  a  moderately  loud  series  of  the 

*  [_E\)  has  for  2nd  partial  crj,  for  3rd  h'ly,  f  [See  App.    XX.  sec.   M.  No.  1,  for  Jen- 

and  hence  for  6th    b'^,  and  for  12th,  h"'<y. —       kin  and  Ewing's  analysis  of  vowel  sounds  by 
Translator. ~\  means  of  the  V\\o\iogvgi\}\i.— Translator.] 


CHAPS.  V.  VI.         APPREHENSION  OF  QUALITIES  OF  TONE.  119 

lower  partial  tones,  up  to  about  the  sixth  partial,  are  nn)rL'  liarinouious  and 
musical.  Compared  with  simple  tones  they  are  rich  and  splendid,  wliilc  they  are 
at  the  same  time  perfectly  sweet  and  soft  if  tlie  higher  \ipper  partials  are  absent. 
To  these  belong  the  musical  tones  j)roduced  by  the  pianoforte,  open  organ  pipes, 
the  softer  piano  tones  of  the  human  voice  and  of  the  French  horn.  The  last- 
named  tones  form  the  transition  to  musical  tones  with  high  upper  partials ;  while 
the  tones  of  flutes,  and  of  pipes  on  the  flue-stops  of  organs  with  a  low  pressure 
of  wind,  ap})roach  to  simple  tones. 

3.  If  only  the  unevenly  numbered  partials  are  present  (as  in  narrow  stopped 
organ  pi])es,  pianoforte  strings  struck  in  their  middle  points,  and  clarinets),  the 
quality  of  tone  is  hoUow,  and,  when  a  large  number  of  such  upper  partials  are 
present,  naml.  When  the  prime  tone  predominates  the  quality  of  tone  is  rich ; 
but  when  the  prime  tone  is  not  sufficiently  superior  in  strength  to  the  upper 
partials,  the  quality  of  tone  is  j^oor.  Thus  the  quality  of  tone  in  the  wider  open  ^ 
organ  pipes  is  richer  than  that  in  the  naiTower :  strings  struck  with  pianoforte 
hammers  give  tones  of  a  richer  quality  than  when  struck  by  a  stick  or  plucked 
by  the  finger ;  the  tones  of  reed  pipes  with  suitable  resonance  chamliers  have  a 
richer  (juality  than  those  without  resonance  chambers. 

4.  When  partial  tones  higher  than  the  sixth  or  seventh  are  very  distinct,  the 
quality  of  tone  is  cutting  and  r<mfjh.  The  reason  for  this  will  be  seen  hei-eafter  to 
lie  in  the  dissonances  which  they  form  with  one  another.  The  degree  of  harshness 
may  be  very  different.  When  their  force  is  inconsiderable  the  higher  upper  [)artials 
do  not  essentially  detract  from  the  nuisical  applicability  of  the  compound  tones ; 
on  the  contrary,  they  ai-e  useful  in  giving  character  and  expression  to  the  music. 
The  most  important  musical  tones  of  this  description  are  those  of  bowx>d  instru- 
ments and  of  most  reed  pipes,  oboe  (hautbois),  bassoon  (fagotto),  harmonium,  and 
the  human  voice.  The  rough,  braying  tones  of  brass  instruments  are  extremely 
penetrating,  and  hence  are  better  adapted  to  give  the  impression  of  great  power  ^ 
than  similar  tones  of  a  softer  iiuality.  They  are  conseciuently  little  suitable  for 
artistic  music  when  used  alone,  but  produce  great  effect  in  an  orchestra.  Why 
high  dissonant  upper  partials  should  make  a  musical  tone  more  penetrating  will 
appear  hereafter. 


CHAPTER  VI. 

ox    THE    APPREHENSION    OF    QUALITIES    OF    TONE. 

Up  to  this  point  we  have  not  endeavoured  to  analyse  given  musical  tones  further 
than  to  determine  the  differences  in  the  number  and  loudness  of  their  partial  tones. 
Before  we  can  determine  the  function  of  the  ear  in  apprehending  qualities  of  tone,  *\ 
we  must  inquire  whether  a  determinate  relative  strength  of  the  upper  partials 
suffices  to  give  us  the  impression  of  a  determinate  musical  quality  of  tone  or 
whether  there  are  not  also  other  perceptible  differences  in  quality  which  are 
independent  of  such  a  relation.  Since  we  deal  only  with  musical  tones,  that  is, 
with  such  as  are  produced  by  exactly  periodic  motions  of  the  air,  and  exclude  all 
irregular  motions  of  the  air  which  appear  as  noises,  we  can  give  this  question  a 
more  definite  form.  If  we  suppose  the  motion  of  the  air  corresponding  to  the 
given  musical  tone  to  be  resolved  into  a  sum  of  pendular  vibrations  of  air,  such 
individual  pendular  vibrations  will  not  only  differ  from  each  other  in  force  or 
amplitude  for  different  forms  of  the  compound  motion,  but  also  in  their  relative 
position,  or,  according  to  physical  terminology,  in  their  difference  of  phase.  For 
example,  if  we  superimpose  the  two  pendular  vibrational  curves  A  and  B,  fig.  31 
(p.  120(t),  first  with  the  point  e  of  B  on  the  point  do  of  A,  and  next  with  the  point 
e  of  B  on  the  point  d^  of  A,  we  obtain  the  two  entirely  distinct  vibrational   curves 


120 


DOES  QUALITY  DEPEND  ON  PHASE? 


PART    I. 


C  and  D.  By  further  displacement  of  the  initial  point  e  so  as  to  place  it  on  dj  or 
dg  we  obtain  other  forms,  which  are  the  inversions  of  the  forms  C  and  D,  as  has 
been  already  shown  (supra,  p.  32a).  If,  then,  musical  quality  of  tone  depends  solely 
on  the  relative  force  of  the  partial  tones,  all  the  various  motions  C,  D,  <fec.,  must 

Fig.  31. 


make  the  same  impression  on  the  ear.      But  if  the  relative  position  of  the  two 
^  waves,  that  is  the  difference  of  phase,  produces  any  effect,  they  must  make  different 
impressions  on  the  ear. 

Now  to  determine  this  point  it  was  necessary  to  compound  various  musical 
tones  out  of  simple  tones  artificially,  and  to  see  whether  an  alteration  of  quality 
ensued  when  force  was  constant  but  phase  varied.  Simple  tones  of  great  purity, 
which  can  have  both  their  force  and  phase  exactly  regulated,  are  best  obtained 
from  tuning-forks  having  the  lowest  proper  tone  reinforced,  as  has  been  already 
described  (p.  54(/ ),  by  a  resonance  chamber,  and  communicated  to  the  air.  To  set 
the  tuning-forks  in  very  uniform  motion,  they  were  placed  between  the  limbs  of  a 
little  electro-magnet,  as  shown  in  fig.  32,  opposite.  Each  tuning-fork  was  screwed 
into  a  separate  board  d  d,  which  rested  upon  pieces  of  india-rubber  tubing  e  e  that 
were  cemented  below  it,  to  prevent  the  vibrations  of  the  fork  from  being  directly 
communicated  to  the  table  and  hence  becoming  audible.  The  limbs  b  b  of  the 
electro-magnet  are  surrounded  with  wire,  and  its  pole  f  is  directed  to  the  fork. 
^  There  are  two  clamp  screws  g  on  the  boai'd  d  d  which  are  in  conductive  connection 
with  the  coils  of  the  electro-magnet,  and  serve  to  introduce  other  wires  which 
conduct  the  electric  current.  To  set  the  forks  in  strong  vibration  the  strength  of 
these  streams  must  alternate  periodically.  These  are  generated  by  a  separate 
apparatus  to  be  presently  described  (fig.  33,  p.  122^,  c). 

When  forks  thus  arranged  are  set  in  vibration,  veiy  little  indeed  of  their  tone 
is  heard,  because  they  have  so  little  means  of  communicating  their  vibrations  to 
the  surrounding  air  or  adjacent  solids.  To  make  the  tone  strongly  audible,  the 
resonance  chamber  i,  which  has  been  previously  tuned  to  the  pitch  of  the  fork, 
must  be  brought  near  it.  This  resonance  chamber  is  fastened  to  another  board  k, 
which  slides  in  a  proper  groove  made  in  the  board  d  d,  and  thus  allows  its  opening 
to  be  brought  very  near  to  the  fork.  In  the  figure  the  resonance  chamber  is  shown 
at  a  distance  from  the  fork  in  order  to  exhibit  the  separate  parts  distinctly ;  when 
in  use,  it  is  brought  as  close  as  possible  to  the  fork.  The  mouth  of  the  resonance 
chamber  can  be  closed  by  a  lid  1  attached  to  a  lever  ra.     By  pulling  the  string  n 


ARTIFICIAL   VOWELS. 


121 


the  lid  is  withdrawn  from  the  opening  and  the  tone  of  the  fork  is  connnunicatod 
to  the  air  with  great  force.  When  the  tlu-ead  is  let  loose,  the  lid  is  brought  over 
the  mouth  of  the  chamber  by  the  sjiring  p,  and  the  tone  of  the  fork  is  no  longer 
heard.  By  partial  opening  of  the  mouth  of  the  chamber,  the  tone  of  the  fork  can 
be  made  to  receive  any  desired  intermediate  degi-ec  of  strength.  The  whole  of 
the  strings  which  open  the  various  resonance  chambers  belonging  to  a  series  of 
such  forks  are  attached  to  a  keyboard  in  such  a  way  that  by  pressing  a  key  the 
corresponding  chamber  is  opened. 

At  first  1  had  eight  forks  of  tliis  kind,  giving  the  tones  B\)  and  its  ilrst  seven 
harmonic  upper  partials,  namely,  l>\f,  ./",  Ij\f,  d" ,  ./",  a"\)*,  and  b"\).  The  prime 
tone  B\f  corresponds  to  the  pitch  in  which  bass  voices  naturally  speak.  Afterwards 
I  had  forks  made  of  the  pitches  d'" ,  ,/"",  a"'\}*  and  //"b,  and  assumed  h\y  for  the 
prime  of  the  compound  tone. 

To  set  the  forks  in  motion,  intermittent  electrical  currents  had  to  be  conducted  ^1 
through  the  coils  of  the  electro-magnet,   giving  as  many  electrical  shocks  as  the 


lowest  forks  made  vibrations  in  a  second,  namely  120.  Every  shock  makes  the 
iron  of  the  electro-niagnet  b  b  momentarily  magnetic,  and  hence  enables  it  to 
attract  the  prongs  of  the  fork,  which  are  themselves  rendered  permanently  magnetic. 
The  prongs  of  the  lowest  fork  ^b  are  thus  attracted  by  the  poles  of  the  electro-  U 
magnet,  for  a  very  short  time,  once  in  every  vibration ;  the  prongs  of  the  second 
for  b\f,  which  moves  twice  as  fast,  once  every  second  vibration,  and  so  on.  The 
vibrations  of  the  forks  are  therefore  both  excited  and  kept  up  as  long  as  the  electric 
currents  pass  through  the  apparatus.  The  vibrations  of  the  lower  forks  are  very 
powerful,  those  of  the  higher  proportionally  weaker. 

The  apparatus  shown  in  fig.  33  (p.  122/.,  c)  serves  to  produce  intermittent  currents 
of  exactly  determinate  periodicity.  A  tuning-fork  a  is  fixed  horizontally  between 
the  limbs  b  b  of  an  electro-magnet ;  at  its  extremities  are  fastened  two  platinum 
wires  c  c,  which  dip  into  two  little  cups  d  filled  half  with  mercury  and  half  with 
alcohol,  forming  the  upper  extremities  of  brass  columns.  These  columns  have  clamp- 
ing screws  i  to  receive  the  wires,  and  stand  on  two  boards  f,  g,  which  turn  about 
an  axis,  as  at  f,  and  wliich  can  each  be  somewhat  raised  or  lowered  by  a  thumb- 

*  [These   being   7th    harmonics    "a"\y   and 
V"|j    are    27    cents     flatter     than    the    «"[j 


and  rt"'t»,  in  the  justly  intoned  scale  of  c\f. 
Translator.'] 


122 


ARTIFICIAL  VOWELS. 


screw,  as  at  g,  so  as  to  make  the  points  of  the  pUxtinum  wires  c  c  exactly  touch 
the  mercury  below  the  alcoliol  in  the  cups  d.  A  third  clamping  screw  e  is  in  con- 
ductive connection  with  the  handle  of  the  tuning-fork.  When  the  fork  vibrates, 
and  an  electric  current  passes  through  it  from  i  to  e,  the  current  will  be  broken 
every  time  that  the  end  of  the  foi-k  a  rises  above  the  surface  of  the  mercury  in  the 
cup  d,  and  re-made  every  time  the  platinum  wire  dips  again  into  the  mercury. 
This  intermittent  current  being  at  the  same  time  conducted  through  the  electro- 
magnet b  b,  fig.  33,  the  latter  becomes  magnetic  every  time  it  passes,  and  thus 
keeps  up  the  vibrations  of  the  fork  a,  which  is  itself  magnetic.  Generally  only 
one  of  the  cups  d  is  used  for  conducting  the  current.  Alcohol  is  poured  over  the 
mercury  to  prevent  the  latter  from  being  burned  by  the  electrical  sparks  which 
arise  when  the  stream  is  interrupted.  This  metliod  of  interrupting  the  current 
was  invented  by  Neef,  who  used  a  simple  vibrating  spring  in  place  of  the  tuning- 
Ufork,  as  may  be  generally  seen  in  the  induction  apparatus  so  much  used  for  medical 
purposes.     But  the  vibrations  of  a  spring  communicate  themselves  to  all  adjacent 


bodies  and  are  for  our  purjjoses  both  too  audible  and  too  irregular.  Hence  the 
necessity  of  substituting  a  tuning-fork  for  the  spring.  The  handle  of  a  well  worked 
symmetrical  tuning-fork  is  extremely  little  agitated  by  the  vibrations  of  the  fork 
and  hence  does  not  itself  agitate  the  bodies  connected  with  it,  so  powerfully  as  the 
II  fixed  end  of  a  straight  spring.  The  tuning-fork  of  the  apparatus  in  fig.  33  must 
be  in  exact  unison  with  the  prime  tone  B\f.  To  effect  this  I  employ  a  little  clamp 
of  thick  steel  wire  h,  placed  on  one  of  the  prongs.  By  slipping  this  towards  the 
free  end  of  the  prong  the  tone  is  deepened,  and  l)y  slipjjing  it  towards  the  handle 
of  the  fork,  the  tone  is  raised.* 

When  the  whole  apparatus  is  in  action,  but  the  resonance  chambers  are  closed, 
all  the  forks  are  maintained  in  a  state  of  uniform  motion,  but  no  sound  is  heai'd, 
beyond  a  gentle  humming  caused  by  the  direct  action  of  the  forks  on  the  air.  But 
on  opening  one  or  more  resonance  chambers,  the  corresponding  tones  are  heard 
with  sufficient  loudness,  and  are  louder  as  the  lid  is  more  widely  opened.  By  this 
means  it  is  possible  to  form,  in  rapid  succession,  different  combinations  of  the  prime 


*  This  apparatus  was  made  by  Fessel  in 
Cologne.  More  detailed  descriptions  of  its 
separate  parts,  and  instructions  for  the  ex- 
periments to  be  made  by  its  means,  are  given 


in  Appendix  VIII.  [This  apparatus  was  ex- 
hibited by  E.  Koenig  (see  Appendix  II.)  in  the 
International  Exhibition  of  1872  in  London. 
— Translator. 1 


CHAP.  VI.  ARTIFICIAL  VOWELS.  123 

tone  with  one  or  more  harmonic  u^jper  partials  havin>f  different  degrees  of  loudness, 
and  thus  produce  tones  of  different  qualities. 

Among  the  natural  musical  tones  which  appear  suitable  for  imitation  with  forks, 
the  vowels  of  the  human  voice  hold  the  first  raidi,  because  they  are  accompanied  by 
comparatively  little  extraneous  noise  and  show  distinct  differences  of  quality  which 
are  easy  to  seize.  Most  vowels  also  are  characterised  by  comparatively  low  upper 
partials,  which  can  be  reached  by  our  forks ;  E  and  I  alone  somewhat  exceed  these 
limits.  The  motion  of  the  very  high  forks  is  too  weak  for  this  purpose  when  in- 
fluenced only  by  such  electrical  currents  as  I  was  able  to  use  without  disturbance 
from  the  noise  of  the  electric  sparks. 

The  first  series  of  experiments  was  made  with  the  eight  forks  B\)  to  b"\f.  With 
these  U,  0,  0,  and  even  A  could  be  imitated  ;  the  last  not  very  well  because  of  my 
not  possessing  the  upper  partials  c"  and  (/'",  which  lie  immediately  above  its 
characteristic  tone  h"\f,  and  are  sensibly  reinforced  in  the  natural  sound  of  this^I 
vowel.  The  prime  tone  B\}  of  this  series,  when  sounded  alone,  gave  a  very  dull 
U,  much  duller  than  could  be  produced  in  speech.  The  sound  became  more  like 
U  when  the  second  and  third  partial  tones  l>\)  and  /'  were  allowed  to  sound  feebly 
at  the  same  time.  A  very  tine  0  was  produced  by  taking  b'\y  strong,  and  h\),  f ,  d" 
more  feebly ;  the  prime  tone  B\)  had  then,  however,  to  be  somewhat  damped.  On 
suddenly  changing  the  pressure  on  the  keys  and  hence  the  position  of  the  lids 
before  the  resonance  chambers,  so  as  to  give  B\)  strong,  and  all  the  upper  partials 
weak,  the  apparatus  uttered  a  good  clear  U  after  the  O. 

A  or  rather  A°  [nearly  o  in  not]  was  produced  by  making  the  fifth  to  the  eighth 
partial  tones  as  loud  as  possible,  and  keeping  the  rest  under. 

The  vowels  of  the  second  and  third  series,  which  have  higher  characteristic  tones, 
could  be  only  imperfectly  imitated  by  bringing  out  their  reinforced  tones  of  the  lower 
])itch.  Though  not  very  clear  in  themselves  they  became  so  by  contrast  on  alterna- 
tion with  U  and  0.  Thus  a  passably  clear  A  was  obtained  by  giving  loudness  H 
chiefly  to  the  fourth  and  fifth  tones,  and  keeping  down  the  lower  ones,  and  a  sort 
of  E  by  reinforcing  the  third,  and  letting  the  rest  sound  feebly.  The  difference 
between  (J  and  these  two  vowels  lay  principally  in  keeping  the  prime  tone  B\)  and 
its  Octave  b\f  much  weaker  for  A  and  E  than  for  0.* 

To  extend  my  experiments  to  the  brighter  vowels,  I  afterwards  added  the  forks 
d"',f"\  a"'\),  b"'\},  the  two  upper  ones  of  which,  however,  gave  a  very  faint  tone, 
and  I  chose  b\y  as  the  prime  tone  in  place  of  B\^.  With  these  I  got  a  very  good  A 
and  A,  and  at  least  a  much  more  distinct  E  than  before.  But  I  could  not  get  up 
to  the  high  characteristic  tone  of  I. 

In  this  higher  series  of  forks,  the  prime  tone  b\),  when  sounded  alone,  repro- 
duced U.  The  same  prime  b\f  with  moderate  force,  accompanied  with  a  strong 
Octave  b'\f,  and  a  weaker  Twelfth/",  gave  0,  which  has  the  characteristic  tone  b'\f. 
A  was  obtained  by  taking  b\f,  b''^,  and/"  moderately^  strong,  and  the  characteristic 
tones  6" jj  and  0?"' very  strong.  To  change  A  into  A  it  was  necessary  to  increase^ 
somewhat  the  force  of  b'\^  and  /"  which  were  adjacent  to  the  characteristic  tone 
d",  to  damp  b"\f,  and  bring  out  (/"'  and/"'  as  strongly  as  possible.  For  E  the  two 
deepest  tones  of  the  series,  b\)  and  b'\),  had  to  be  kept  moderately  loud,  as  being 
adjacent  to  the  deeper  characteristic  tone/',  whilst  the  highest/'",  a"'[>,  b"'\y  had 
to  be  made  as  prominent  as  possible.  But  I  have  hitherto  not  succeeded  so  well 
with  this  as  with  the  other  vowels,  because  the  high  forks  were  too  weak,  and 
because  perhaps  the  upper  partials  which  lie  above  the  characteristic  tone  b"'\) 
could  not  be  entirely  dispensed  with.t 

*  The  statements  iu  the  Miincheiier  (jelchrte  above  results  will  serve  to  show  tlieir  relations 

Anzcujcn  for  June  20,  1859,  must  be  corrected  more  clearly.     In  the  first  line  arc  placed  the 

accordingly.     At  that  time  I  did  not  know  the  notes  of  the  forks  and  the  numbers  of  the 

higher  upper  partials  of  E  and  I,  and  hence  corresponding  partials.     The  letters  ^V''- ?'.  "/. 

made  the  O  too  dull  to  distinguish  it  from  the  /,  ff  below  them  are  the  usual  musical  indica- 

imperfect  E.  tions  of  force,  p('«7i/6s/wio,  piano,  mezzo  forte, 

t  [The  following  tabular  statement  of  the  forte,  fortissimo.       Where   no    such   mark   is 


124 


QUALITY  INDEPENDENT  OV  PHASE. 


In  precisely  the  same  way  as  the  vowels  of  the  human  voice,  it  is  possible  to 
imitate  the  quality  of  tone  produced  by  organ  pipes  of  different  stops,  if  they  have 
not  secondary  tones  which  are  too  high,  but  of  course  the  whizzing  noise,  formed 
by  breaking  the  stream  of  air  at  the  lip,  is  wanting  in  these  imitations.  The 
tuning-forks  are  necessarily  limited  to  the  imitation  of  the  purely  musical  part  of 
the  tone.  The  piercing  high  upper  partials,  required  for  imitating  reed  instru- 
ments, were  absent,  but  the  nasality  of  the  clarinet  was  given  by  using  a  series 
of  unevenly  numbered  partials,  and  the  softer  tones  of  the  horn  by  the  full  chorus 
of  all  the  forks. 

But  though  it  was  not  possible  to  imitate  every  kind  of  quality  of  tone  by  the 
present  apparatus,  it  sufficed  to  decide  the  important  question  as  to  the  effect  of 
altered  difference  of  phase  upon  quality  of  tone.  As  I  particularly  observed  at  the 
beginning  of  this  chapter,  this  question  is  of  fundamental  importance  for  the 
H  theory  of  auditory  sensation.  The  reader  who  is  unused  to  physical  investigations 
must  excuse  some  apparently  difiicidt  and  dry  details  in  the  explanation  of  the 
experiments  necessary  for  its  decision. 

The  simple  means  of  altering  the  phases  of  the  secondary  tones  consists  in 
bringing  the  resonance  chambers  somewhat  out  of  tune  by  narrowing  their 
apertures,  which  weakens  the  resonance,  and  at  the  same  time  alters  the  phase. 
If  the  resonance  chamber  is  tuned  so  that  the  simple  tone  which  excites  its 
strongest  resonance  coincides  with  the  simple  tone  of  the  corresponding  fork,  then, 
as  the  mathematical  theory  shows,*  the  greatest  velocity  of  the  air  at  the  mouth 
of  the  chamber  in  an  outward  direction,  coincides  with  the  greatest  velocity  of  the 
ends  of  the  fork  in  an  inward  direction.  On  the  other  hand,  if  the  chamber  is 
tuned  to  be  slightly  deeper  than  the  fork,  the  greatest  velocity  of  the  air  slightly 
])recedes,  and  if  it  is  tuned  slightly  higher,  that  greatest  velocity  slightly  lags 
behind  the  greatest  velocity  of  the  fork.  The  more  the  tuning  is  altered,  the 
^greater  will  be  the  difference  of  phase,  till  at  last  it  reaches  the  duration  of  a 
quarter  of  a  vibration.  The  magnitude  of  the  difference  of  phase  agrees  during 
this  change  precisely  with  the  strength  of  the  resonance,  so  that  to  a  certain  degree 
we  are  able  to  measure  the  former  by  the  latter.  If  we  represent  the  strength  of 
the  sound  in  the  resonance  chamber  when  in  unison  with  the  fork  by  10,  and 
divide  the  periodic  time  of  a  vibration,  like  the  circumference  of  a  circle,  into  360 


added  the  partial  is  not  mentioned  in  the  text.       ones,  but  the  whole 
For  the  second  series  of  experiments  the  forks       tials  of  b\f. 
of  cori'csponding  pitches  are  kept  under  the  old 


now  numbered  as  par- 


11       % 


First  ) 
Forks  )■ 

1 

2 

3       '       4 

5_ 

6 

7 

«"b 

8 

10 

12 

14 

16 

% 

U 
0 

A 

E 

/ 

■'»/ 
p 
p 
p 

PP 
P 
P 
PP 
PP 

PP 

P      1      / 
P      I      P 
P      \      f 
f            P 

/ 
P 

ff 
P 

ff 
P 

ff 
P 

Second  ( 
Forks    r 

1 

•2 

3 

4 

5 

6 

7 

8 

1 

> 

U 
0 

A 
E 

if 

mf 

J' 

mf 
mf 

P 

mf 
f 

/ 
P 

f 

ff 

ff 
ff 

ff 

ff 

See  Appendix  XX.  sect.  !M.  No.  2,  for 
Messrs.  Preece  and  Stroh's  new  method  of 
vowel  synthesis. —  Tr(mshitor.~\ 


See  the  first  part  of  Appendix  IX. 


QUALITY  INDEPENDENT  OF  PHASE. 


125 


degrees,  the  relation  between  the  strength  of  the  resonance  and  the  diti'erence  of 
phase  is  shown  by  the  following  table  : — 


strength  of 

Difference  of  Phase  i 

1  angular 

Resonance. 

degrees. 

10 

0- 

9 

35°  54' 

8 

50°  12' 

7 

60°  40' 

6 

68°  54' 

5 

75°  31' 

4 

80°  48' 

3 

84°  50' 

2 

87°  42' 

1 

89°  26' 

u 

This  table  shows  that  a  comparatively  slight  weakening  of  resonance  by 
altering  the  tuning  of  the  chamber  occasions  considerable  differences  of  phase, 
but  that  when  the  weakening  is  considerable  there  are  relatively  slight  changes 
of  phase.  We  can  take  advantage  of  this  circumstance  when  compounding  the 
vowel  sounds  by  means  of  the  tuning-forks  to  produce  every  possible  alteration  of 
phase.  It  is  only  necessary  to  let  the  lid  shade  the  mouth  of  the  resonance 
chamber  till  the  strength  of  the  tone  is  perceptibly  diminished.  As  soon  as  we 
have  learned  how  to  estimate  roughly  the  amount  of  diminution  of  loudness,  the 
.above  table  gives  us  the  corresponding  alteration  of  phase.  We  are  thus  able  to 
alter  the  vibrations  of  the  tones  in  question  to  any  amount,  up  to  a  quarter  of  the 
periodic  time  of  a  vibration.  Alterations  of  phase  to  the  amount  of  half  the 
periodic  time  are  produced  by  sending  the  electric  current  through  the  electro- 
magnets of  the  corresponding  fork  in  an  opposite  direction,  which  causes  the  ends 
of  the  fork  to  be  repelled  instead  of  attracted  by  the  electro-magnets  on  the  H 
passage  of  the  current,  and  thus  sets  the  fork  vibrating  in  the  contrary  direction. 
This  counter-excitement  of  the  fork,  however,  by  repelling  currents,  must  not  be 
continued  too  long,  as  the  magnetism  of  the  fork  itself  wovild  otherwise  gradually 
diminish,  whereas  attracting  currents  strengthen  it  or  maintain  it  at  a  maximum. 
It  is  well  known  that  the  magnetism  of  masses  of  iron  that  are  violently  agitated 
is  easily  altered. 

After  a  tone  has  been  compounded,  in  which  some  of  the  partials  have  been 
weakened  and  at  the  same  time  altered  in  phase  by  the  half-shading  of  the 
apertures  of  their  corresponding  resonance  chambers,  we  can  re-compound  the 
same  tone  by  an  equal  amount  of  weakening  in  the  same  partials,  but  without 
shading  the  aperture,  and  thei-efore  without  change  of  phase,  by  simply  leaving 
the  mouths  of  the  chambers  wide  open,  and  increasing  their  distances  from  the 
exciting  forks,  until  the  required  amount  of  enfeeblement  of  sound  is  attained. 

For  example,  let  us  first  sound  the  forks  B\)  and  h\},  with  fully  opened  resonance  H 
chambers,  and  perfect  accord.  They  will  vibrate  as  shewn  by  the  vibrational 
forms  fig.  31,  A  and  B  (p.  120rt),  with  the  points  e  and  do  coincident,  and  produce 
at  a  distance  the  compound  vibration  represented  by  the  vibrational  curve  C.  But 
by  closing  the  resonance  chamber  of  the  fork  B\)  we  can  make  the  point  e  on  the 
curve  B  coincide  with  the  points  between  d„  and  di  on  the  curve  A.  To  make  e 
coincide  with  dj,  the  loudness  of  B\)  must  be  made  about  three-quarters  of  what 
it  would  be  if  the  mouth  of  the  chamber  were  unshaded.  The  point  e  can  be  made 
to  coincide  with  d4  by  reversing  the  current  in  the  electro-magnets  and  fully 
opening  the  mouth  of  the  resonance  chamber ;  and  then  by  imperfectly  opening 
the  chamber  of  B\)  the  point  e  can  be  made  to  move  towards  S.  On  the  other 
hand,  an  imperfect  opening  of  the  chamber  b\)  will  make  e  recede  from  coincidence 
with  S  (which  is  the  same  thing  as  coincidence  with  do)  or  with  dj,  towards  d^  or 
da  respectively.     The  proportions  of  loudness  may  be  made  the  same  in  all  these 


126  QUALITY  INDEPENDENT  OF  PHASE.  part  i. 

cases,  without  any  alteration  of  pliase,  by  removing  the  corresponding  chambers  to 
the  proper  distance  from  its  forks  without  shading  its  mouth. 

In  this  manner  every  possible  difference  of  phase  in  the  tones  of  two  chambers 
can  be  produced.  The  same  process  can  of  course  be  applied  to  any  required 
number  of  forks.  I  have  thus  experimented  upon  numerous  combinations  of  tone 
with  varied  differences  of  phase,  and  I  have  never  experienced  the  slightest  dif- 
ference in  the  quality  of  tone.  So  far  as  the  quality  of  tone  was  concerned,  I 
found  that  it  was  entirely  indiffei-ent  whether  I  weakened  the  separate  partial 
tones  by  shading  the  moutlis  of  their  resonance  chambers,  or  by  moA'ing  the 
chamber  itself  to  a  sufficient  distance  from  the  fork.  Hence  the  answer  to  the 
proposed  question  is  :  the  qxiality  of  the  musical  j^ortion  of  a  compound  tone  depends 
solely  on  the  number  aiul  relative  strength  of  its  partial  simple  tones,  and  in  no  respect 
on  their  differences  of  2'>hase* 

The  preceding  proof  that  quality  of  tone  is  independent  of  difference  of 
phase,  is  the  easiest  to  carry  out  experimentally,  but  its  force  lies  solely  in  the 
theoretical  proposition  that  phases  alter  contemporaneously  with  strength  of  tone 
when  the  mouths  of  the  resonance  chambers  are  shaded,  and  this  proposition  is 
the  result  of  mathematical  theory  alone.  We  cannot  make  vibrations  of  air 
directly  visible.  But  by  a  slight  change  in  the  experiment  it  may  be  so  conducted 
as  to  make  the  alteration  of  phase  immediately  visible.  It  is  only  necessary  to 
put  the  tuning-forks  themselves  out  of  tune  with  their  resonance  chambers,  by 
attaching  little  lumps  of  wax  to  the  prongs.  The  same  law  holds  for  the  phases 
of  a  tuning-fork  kept  in  vibration  by  an  electric  current,  as  for  the  resonance 
chambers  themselves.  The  phase  gradually  alters  by  a  quarter  period,  while  the 
strength  of  the  tone  of  the  fork  is  reduced  from  a  maximum  to  nothing  at  all,  by 
putting  it  out  of  time.  The  phase  of  the  motion  of  the  air  retains  the  same 
relation  to  the  phase  of  the  vibration  of  the  fork,  because  the  pitch,  which  is 
determined  by  the  number  of  interruptions  of  the  electrical  current  in  a  second,  is 
not  altered  by  the  alteration  of  the  fork.  The  change  of  phase  in  the  fork  can  be 
observed  directly  by  means  of  Lissajou's  vibration  microscope,  already  described 
and  shown  in  fig.  22  (p.  80(i).  Place  the  prongs  of  the  fork  and  the  microscope  of 
this  instrument  horizontally,  and  the  fork  to  be  examined  vertically ;  powder  the 
upper  end  of  one  of  its  prongs  with  a  little  starch,  direct  the  microscope  to  one  of 
the  grains  of  starch,  and  excite  both  forks  by  means  of  the  electrical  currents  of 
the  interrupting  fork  (fig.  33,  p.  \'22h).  The  fork  of  Lissajou's  instrument  is  in 
unison  with  the  interrupting  fork.  The  grain  of  starch  vibrates  horizontally,  the 
object-glass  of  the  microscope  vertically,  and  thus,  by  the  composition  of  these 
two  motions,  curves  are  generated,  just  as  in  the  observations  on  violin  strings 
previously  described. 

When  the  observed  fork  is  in  unison    with    the   interrupting  fork,   the  curve 
becomes   an  oblique  straight  line  (fig.   34,    1),    if   both  forks  pass    through    their 

Fig.  34. 


position  of  rest  at  the  same  moment.  As  the  phase  alters,  the  straight  line  })aases 
thi-ough  a  long  oblique  ellipse  (2,  3),  till  on  the  difference  of  phase  becoming  a 
quarter  of  a  period,  it  develops  into  a  circle  (4) ;  and  then  as  the  difference  of 
phase  increases,  it  passes  through  oblique  ellipses  (5,  6)  in  another  direction,  till  it 
reaches  another  straight  line  (7),  on  the  difference  becoming  half  a  period. 

If  the  second  fork   is  the  upper  Octave  of  the  inten'upting  fork,  the  curves 

*  [The  experiments  of  Koenig  with  the  modification.  Moreover  Koenig  contends  that 
wave-siren,  explained  in  App.  XX.  sect.  L.  the  'apparent  exception'  of  p.  127c,  is  an 
art.  6,   show  that  this  law  requires  a  slight       '  actual '  one  {ibid.). — Trmislator,] 


CHAP.  VI.  QUALITY  INDEPENDENT  OF  PHASE.  127 

1,  2,  .3,  4,  5,  in  fig.  35,  show  tlie  series  of  forms.  Here  3  answers  to  the  case  when 
both  forks  pass  through  tlieir  position  of  rest  at  the  same  time ;  2  and  4  diflter  from 
that  position  by  yV,  and  1  and  5  by  j  of  a  wave  of  the  higher  fork. 

If  we  now  bring  the   forks   into  the   most  perfect    possible    unison   with    the 
interrupting  fork,  so  that  both  vibrate  as  strongly  as  possible,  and  then  alter  their 

Fig.  35. 
1  _  2  „    „       3 


tuning  v.  little  by  putting  on  or  removing  pieces  of  wax,  we  also  see  one  figure  of  the 
microscopic  image  gradually  passing  into  another,  and  can  thus  easily  assure  our-  H 
selves  of  the  correctness  of  the  law  already  cited.  Experiments  on  quality  of  tone 
are  then  conducted  by  first  bringing  all  the  forks  as  exactly  as  possible  to  the 
pitches  of  the  hai-monic  upper  partial  tones  of  the  interrupting  fork,  next  removing 
the  resonance  chambers  to  such  distances  from  the  forks  as  will  give  the  required 
relations  of  strength,  and  finally  putting  the  forks  out  of  tune  as  much  as  we  please 
by  sticking  on  lumps  of  wax.  The  size  of  these  lumps  should  be  previously  so 
regulated  by  microscopical  observation  as  to  produce  the  required  difl^erence  of 
phase.  This,  however,  at  the  same  time  weakens  the  vibrations  of  the  forks,  and 
hence  the  strength  of  the  tones  must  be  restored  to  its  former  state  by  bringing  the 
resonance  chambers  nearer  to  the  forks. 

The  result  in  these  experiments,  where  the  forks  are  put  out  of  tune,  is  the 
same  as  in  those  where  the  resonance  chambers  were  put  out  of  tune.  There  is 
no  perceptible  alteration  of  quality  of  tone.  At  least  there  is  no  alteration  so 
marked  as  to  be  recognisable  after  the  expiration  of  the  few  seconds  necessary  % 
for  resetting  the  apparatus,  and  hence  certainly  no  such  change  of  quality  as 
would  change  one  vowel  into  another. 

An  apparent  exception  to  this  rule  must  here  be  mentioned.  If  the  forks  B\} 
and  h\}  are  not  perfectly  tuned  as  Octaves,  and  are  brought  into  vibration  by  rub- 
bing or  striking,  an  attentive  ear  will  observe  very  weak  beats  which  appear  like 
small  changes  in  the  strength  of  the  tone  and  its  quality.  These  beats  are  cer- 
tainly connected  with  the  successive  entrance  of  the  vibrating  forks  on  varying 
difference  of  phase.  Their  explanation  will  be  given  when  combinational  tones  are 
considered,  and  it  will  then  be  shown  that  these  slight  variations  of  quality  are 
referable  to  changes  in  the  strength  of  one  of  the  simple  tones. 

Hence  we  are  able  to  lay  down  the  important  law  that  differences  iv  musical 
qiudity  of  tone  depend,  solely  on  the  presence  and  strength  of  partial  tones,  and  in 
no  respect  on  the  differences  in  pihase  under  which  these  piartial  tones  enter  into 
composition.  It  must  be  here  observed  that  we  are  speaking  only  of  musical  H 
quality  as  previously  defined.  When  the  musical  tone  is  accompanied  by  un- 
musical noises,  such  as  jarring,  scratching,  soughing,  whizzing,  hissing,  these 
motions  are  either  not  to  be  considered  as  periodic  at  all,  or  else  correspond  to 
high  upper  partials,  of  nearly  the  same  pitch,  which  consequently  form  strident 
dissonances.  We  were  not  able  to  embrace  these  in  our  experiments,  and  hence 
we  must  leave  it  for  the  present  doubtful  whether  in  such  dissonating  tones 
diftereuce  of  phase  is  an  element  of  importance.  Subsequent  theoretic  considera- 
tions will  lead  us  to  suppose  that  it  really  is. 

If  we  wish  only  to  imitate  vowels  by  compound  tones  without  being  al)le  to 
distinguish  the  differences  of  phase  in  the  individual  constituent  simple  tones,  we 
can  effect  our  purpose  tolerably  well  with  organ  pipes.  But  we  must  have  at  least 
two  series  of  them,  loud  open  and  soft  stopped  pipes,  because  the  strength  of  tone 
cannot  be  increased  by  additional  pressiu-e  of  wind  without  at  the  same  time 
changing  the  pitch.     I  have  had  a  double  row  of  pipes  of  this  kind  made  by  Herr 


128  APPREHENSION  OF  QUALITY  OF  TONE.  part  r. 

Appunn  in  Ilanau,  giving  the  first  sixteen  pai-tial  tones  of  B\}.  All  these  pipes 
stand  on  a  common  windchest,  which  also  contains  the  valves  by  which  they  can 
be  opened  or  shut.  Two  larger  valves  cut  off  the  passage  from  the  windchest  to 
the  bellows.  While  these  valves  are  closed,  the  pipe  valves  are  arranged  for  the 
required  combination  of  tones,  and  then  one  of  the  main  valves  of  the  windchest 
is  opened,  allowing  all  the  pipes  to  sound  at  once.  The  character  of  the  vowel  is 
better  produced  in  this  way  by  short  jerks  of  sound,  than  by  a  long  continued 
sovuid.  It  is  best  to  produce  the  prime  tone  and  the  predominant  upper  partial 
tones  of  the  required  vowels  on  both  the  open  and  stopped  pipes  at  once,  and  to 
open  only  the  weak  stopped  pipes  for  the  next  adjacent  tones,  so  that  the  strong- 
tone  may  not  be  too  isolated. 

The  imitation  of  the  vowels  by  this  means  is  not  very  perfect,  because,  among 
other  reasons,  it  is  impossible  to  graduate  the  strength  of  tone  on  the  different  pipes 

IT  80  delicately  as  on  the  tuning-forks,  and  the  higher  tones  especially  are  too  screaming. 
But  the  vowel  sounds  thus  composed  are  perfectly  recognisable. 

We  proceed  now  to  consider  the  part  played  by  the  ear  in  the  apprehension  of 
quality  of  tone.  The  assumption  formerly  made  respecting  the  function  of  the  ear, 
was  that  it  was  capable  of  distinguishing  both  the  pitch  number  of  a  musical  tone 
(which  gives  the  pitch),  and  also  tlie  form  of  the  vibrations  (on  which  the  difference 
of  quality  depends).  This  last  assertion  was  based  simply  on  the  exclusion  of  all 
other  possible  assumptions.  As  it  could  be  proved  that  sameness  of  pitch  always 
required  equal  pitch  numbers,  and  as  loudness  visibly  depended  upon  the  ampli- 
tude of  the  vibrations,  the  quality  of  tone  must  necessarily  depend  on  something 
which  was  neither  the  number  nor  the  amplitude  of  the  vibrations.  There  was 
nothing  left  us  but  form.  We  can  now  make  this  view  more  definite.  The  ex- 
periments just  described  show  that  waves  of  very  different  forms  (as  fig.  31, 
C,  D,  p.  120a,  and  fig.  12,  C,  D,  p.  22/^),  may  have  the  same  quality  of  tone,  and 

U  indeed,  for  every  case,  except  the  simple  tone,  thei-e  is  an  infinite  number  of  forms 
of  wave  of  this  kind,  because  any  alteration  of  the  difference  of  phase  alters  the 
form  of  wave  without  changing  the  quality  of  tone.  The  only  decisive  character 
of  a  quality  of  tone,  is  that  the  motion  of  the  air  which  strikes  the  ear  when  re- 
solved into  a  sum  of  pendulum  vibrations  gives  the  same  degree  of  strength  to  the 
same  simple  vibration. 

Hence  the  ear  does  not  distinguish  the  different  forms  of  waves  in  themselves, 
as  the  eye  distinguishes  the  different  vibrational  curves.  The  ear  must  be  said 
rather  to  decompose  every  wave  form  into  simpler  elements  according  to  a  definite 
law.  It  then  receives  a  sensation  from  each  of  these  simpler  elements  as  from  an 
harmonious  tone.  By  trained  attention  the  ear  is  able  to  become  conscious  of  each 
of  these  simpler  tones  separately.  And  what  the  ear  distinguishes  as  different 
qualities  of  tone  are  only  different  combinations  of  these  simpler  sensations. 

The   comparison   between   ear  and   eye  is  here   very  instructive.       When    the 

H  vibrational  motion  is  rendered  visible,  as  in  the  vibration  microscope,  the  eye  is 
capable  of  distinguishing  every  possible  different  form  of  vibration  one  from 
another,  even  such  as  the  ear  cannot  distinguish.  But  the  eye  is  not  capable  of 
'  directly  resolving  the  vibrations  hito  simple  vibrations,  as  the  ear  is.  Hence  the 
eye,  assisted  by  the  above-named  instrument,  really  distinguishes  the  form  of  vibra- 
tion, as  such,  and  in  so  doing  distinguishes  every  different  form  of  vibration.  The 
ear,  on  the  other  hand,  does  not  distinguish  every  different  form  of  vibration,  but 
only  such  as  when  resolved  into  pendular  vibrations,  give  different  constituents. 
But  on  the  other  hand,  by  its  capability  of  distinguishing  and  feeling  these  very 
constituents,  it  is  again  superior  to  the  eye,  which  is  quite  incapable  of  so  doing. 

This  analysis  of  compound  into  simple  pendular  vibrations  is  an  astonishing 
property  of  the  ear.  The  reader  must  bear  in  mind  that  when  we  apply  the  term 
'  compound '  to  the  vibrations  produced  by  a  single  musical  instrument,  the  '  com- 
position '  has  no  existence  except  for  our  auditory  perceptions,  or  for  mathematical 
theory.     In  reality,  the  motion  of  the  particles  of  the  air  is  not  at  all  compound, 


SYMPATHETICALLY  YIBRATINd  PARTS  OF  THE  EAR. 


129 


it  is  quite  simple,  flowing  from  a  single  source.  When  we  turn  to  external  nature 
for  an  analogue  of  such  an  analysis  of  periodical  motions  into  simple  motions,  we 
find  none  but  the  phenomena  of  sympathetic  vibration.  In  reality  if  we  suppose 
the  dampers  of  a  pianoforte  to  be  raised,  and  allow  any  musical  tone  to  impinge 
powerfully  on  its  sounding  board,  we  bring  a  set  of  strings  into  sympathetic  vibra- 
tion, namely  all  those  strings,  and  only  those,  which  correspond  with  the  simple 
tones  contained  in  the  given  musical  tone.  Here,  then,  we  have,  by  a  purely  me- 
chanical process,  a  resolution  of  air  waves  precisely  similar  to  that  performed  by  the 
ear.  The  air  wave,  quite  simple  in  itself,  brings  a  certain  number  of  strings  into 
sympathetic  vibration,  and  the  sympathetic  vibration  of  these  strings  depends  on 
the  same  law  as  the  sensation  of  harmonic  upper  partial  tones  in  the  ear.* 

There  is  necessarily  a  certain  difference  between  the  two  kinds  of  apparatus, 
because  the  pianoforte  strings  readily  vibrate  with  their  upper  partials  in  sympathy, 
and  hence  separate  into  several  vibrating  sections.  We  will  disregard  this  pecu-  % 
liarity  in  making  our  comparison.  It  would  besides  be  easy  to  make  an  instrument 
in  which  the  strings  would  not  vibrate  sensibly  or  powerfully  for  any  but  their 
prime  tones,  by  simply  loading  the  strings  slightly  in  the  middle.  This  would  make 
their  higher  proper  tones  inharmonic  to  their  primes. 

Now  suppose  we  were  able  to  connect  every  string  of  a  piano  with  a  nervous  fibre 
in  such  a  manner  that  this  fibre  would  be  excited  and  experience  a  sensation  every 
time  the  string  vibrated.  Then  every  musical  tone  which  impinged  on  the  instru- 
ment would  excite,  as  we  know  to  be  really  the  case  in  the  eai-,  a  series  of  sensa- 
tions exactly  corresponding  to  the  pendular  vibrations  into  which  the  original 
motion  of  the  air  had  to  be  resolved.  By  this  means,  then,  the  existence  of  each 
partial  tone  would  be  exactly  so  perceived,  as  it  really  is  perceived  by  the  ear. 
The  sensations  of  simple  tones  of  different  pitch  would  under  the  supposed  con- 
ditions fall  to  the  lot  of  different  nervous  fibres,  and  hence  be  produced  quite 
separately,  and  independently  of  each  other.  ^ 

Now,  as  a  matter  of  fact,  later  microscopic  discoveries  respecting  the  internal 
construction  of  the  ear,  lead  to  the  hypothesis,  that  arrangements  exist  in  the  ear 

similar    to    those    which    we 
^^^"  ^^''  have  imagined.     The  end  of 

every  fibre  of  the  auditory 
nerve  is  connected  with  small 
elastic  parts,  which  we  cannot 
but  assume  to  be  set  in  sym- 
pathetic vibration  by  the 
waves  of  sound. 

The    construction    of   the 
ear  may  be  briefly  described 
as    follows  : — The    fine    ends 
of  the  fibres  of  the  auditory  5^ 
nerves  are  expanded  on  a  deli- 
cate   membrane   in    a    cavity 
filled   with    fluid.     Owing   to 
its  involved  form  this  cavity 
is  known  as  the  lahyrinth  of  the  ear.     To  conduct  the  vibrations  of  the  air  with 
sufficient  force  into  the  fluid  of  the  labyrinth    is  the  office  of  a  second  portion  of 
the    ear,    the    tympdnunn  or   drum  and  the  parts  within  it.     Fig.   36  above  is   a 


*  [Eaise  the  dampers  of  a  piano,  and  utter 
the  vowel  A  («A)  sharply  and  loudly,  directing  it 
well  on  to  the  sound  l)oard,  pause  a  second  and 
the  vowel  will  be  echoed  from  the  strings.  Re- 
damp,  raise  the  dampers  and  cry  U  (00)  as  be- 
fore, and  that  will  also  be  echoed.  Re-damp, 
raise  the  dampers  and  cry  I  (cc),  and  that 
again  will  be  echoed.  The  other  vowels  may 
be  tried  in  the  same  way.     The  echo,  though 


imperfect,  is  always  true  enough  to  surprise 
a  hearer  to  whom  it  is  new,  even  if  the  pitch  of 
the  vowel  is  taken  at  hazard.  It  will  be  im- 
proved if  the  vowels  are  sung  loudly  to  notes 
of  the  piano.  The  experiment  is  so  easy  to 
make  and  so  fundainental  in  character,  that 
it  should  be  witnessed  by  e%'ery  student. — 
Tramlator.^ 


130 


SYMPATHETICALLY  VIBRATING  PARTS  OF  THE  EAR. 


diagrammatic  section,  of  the  size  of  life,  showing  the  cavities  belonging  to  the 
auditory  apparatus.  A  is  the  labyrinth,  B  B  the  cavity  of  the  tynvpdmim  or  drum, 
D  the  funnel-shaped  entrance  into  the  meatus  or  external  auditory  passage,  nar- 
rowest in  the  middle  and  expanding  slightly  towards  its  upper  extremity.  This 
meatus,  in  the  ear  or  passage,  is  a  tube  formed  partly  of  cartilage  or  gristle  and 
partly  of  bone,  and  it  is  separated  from  the  tympanum  or  drum,  by  a  thin  circular 
membrane,  the  memhrana  tympdnl  or  drumshin*  c  c,  which  is  rather  laxly  stretched 
in  a  bony  ring.  The  drum  (tympanum)  B  lies  between  the  outer  passage 
(meatus)  and  the  labyrinth.  The  drum  is  separated  from  the  labyrinth  by  bony 
walls,  pierced  with  two  holes,  closed  by  membranes.  These  are  the  so-called 
windows  {fenes'trae)  of  the  labyrinth.  The  upper  one  o,  called  the  oval  window 
{fenestra  uvdlis),  is  connected  with  one  of  the  ossicles  or  little  bones  of  the  ear 
called    the    stirrup.     The   lower   or   round  window    r    {fenestra    rotunda)    has   no 

H  connection  with  these  ossicles. 

The  drum  of  the  ear  is  consequently  completely  shut  off  from  the  external 
passage  and  from  the  labyrinth.  But  it  has  free  access  to  the  upper  part  of  the 
pharynx  or  throat,  through  the  so-called  Eustachian  t  tube  E,  ^vhich  in  Germany 
is  tenned  a  trumpet,  because  of  the  trumpet-like  expansion  of  its  pharyngeal 
extremity  and  the  narrowness  of  its  opening  into  the  drum.  The  end  which  opens 
into  the  drum  is  formed  of  bone,  but  the  expanded  pharyngeal  end  is  formed  of  thin 
flexible  cartilage  or  gristle,  split  along  its  upper  side.  The  edges  of  the  split  are 
closed  by  a  sinewy  membrane.  By  closing  the  nose  and  mouth,  and  either  con- 
densing the  air  in  the  mouth  by  pressure,  or  rarefying  it  by  suction,  air  can  be 
respectively  driven  into  or  drawn  out  of  the  drum  through  this  tube.  At  the 
entrance  of  air  into  the  drum,  or  its  departure  from  it,  we  feel  a  sudden  jerk  in 
the  ear,  and  hear  a  dull  crack.  Air  passes  from  the  pharynx  to  the  drum,  or  from 
the  drum  to  the  pharynx  only  at  the  moment  of  making  the  motion  of  swallowing. 

H  When  the  air  has  entered  the  drum  it  remains  there,  even  after  nose  and  mouth 
are  opened  again,  until  we  make  another  motion  fig.  37. 

of  swallowing.  Then  the  air  leaves  the  drum, 
as  we  perceive  by  a  second  cracking  in  the  ear, 
and  the  cessation  of  the  feeling  of  tension  in  the 
drumskin  which  had  remained  up  till  that  time. 
These  experiments  shew  that  the  tube  is  not 
usually  open,  but  is  opened  only  diu-ing  swallow- 
ing, and  this  is  explained  by  the  fact  that  the 
muscles  which  raise  the  velum  j^c^''^^  o^  soft 
palate,  and  are  set  in  action  on  swallowing,  arise 
partly  from  the  cartilaginous  extremity  of  the  tube. 
Hence  the  drum  is  generally  quite  closed,  and 
filled   with   air,   which    has   a    pressure    equal   to 

Hthat  of  the  external  air,  because  it  has  from 
time  to  time,  that  is  whenever  we  swallow 
means  of  equalising  itself  with  the  same  by  free 
communication.  For  a  strong  pressure  of  the 
air,  the  tube  opens  even  without  the  action  of 
swallowing,  and  its  power  of  resistance  seems  to 
be  very  different  in  different  individuals. 

In  two  places,  this  air  in  the  drum  is  like- 
wise separated  from  the  fluid  of  the  labj-rinth 
merely   by  a    thin    stretched    membrane,   which    closes    the    two    ivindows   of  the 


the  Ossicles  of  the  ear  in  mutual  connection 
seen  from  the  front,  and  taken  from  the 
right  side  of  the  head,  which  has  been 
turned  a  little  to  the  right  round  a 
vertical  axis.  M  hammer  or  malleus. 
J  anvil  or  incvs.  S  stirrup  or  stapes. 
Mcp  head,  Mc  neck,  Ml  long  process  or 
processus  (jrd'ciiis.  Mm  handle  or  manu- 
brima  of  tlie  hammer. — Jc  body,  Jb  short 
process,  Jl  long  process,  Jpl  orbicular 
process  or  os  orbicXddrc  or  proces'sus 
lentictitarls,  of  the  an\-il.— Sep  head  or 
ccpifuiutn  of  the  stirrup. 


*  [In  conunon  parlance  the  dnanskin  of 
the  ear,  or  tijmpnnic  membrane,  is  spoken  of 
as  the  drum  itself.  Anatomists  as  well  as 
drummers  distinguish  the  membranous  cover 
(drumskin)  which  is  struck,  from  the  hollow 
cavity  (drum)  which  contains  the  resonant  air. 


The  quantities  of  the  Latin  words  are  marked, 
as  I  have  heard  musicians  give  them  incor- 
rectly. —  Translator.  ] 

t  [Generally     pronounced     yoo-stai'-kl-an, 
but  sometimes  yoo-stdi'-shl-an. —  Translator.] 


CHAP.  VI.     8YMPxVTHI^:ti("ally  vii!R.\'n.\(;  I'Airrs  of  thk  km;. 


131 


labyrinth,  already  mentioned,  namely,  the  ond  window  (o,  tiii'.  .'K!,  p.  129c)  and 
the  ronnd  window  (r).  Hotli  of  these  membranes  an'  in  contact  on  their  onter 
side  with  the  air  of  the  drum,  and  on  their  inner  side  with  tlie  water  of  the  laby- 
rinth. Tlie  membrane  of  tlie  ronnd  window  is  free,  b\xt  that  of  tlie  oval  window- 
is  connected  with  the  drumskin  of  the  ear  by  a  sei'ies  of  three  little  bones  or 
auditory  ossicles,  jointed  together.  Fig.  37  shews  the  three  ossicles  in  their  natural 
connection,  enlarged  four  diameters,  'rhey  are  the  ImnDiier  (mal'lcus)  M,  the  unv'd 
(in'cus)  J,  and  the  stirrup  (stapes*)  8.  The  hammer  is  attached  to  the  drumskin, 
and  the  stirrup  to  the  membrane  of  the  oval  window. 

The  hammer  shewn  separately  in  fig.  38,  has  a  thick,  rounded  npi)er  extremity, 

the  head  cp,  and  a   thinner  lower  extremity,  tlie  iKUKlIf  m.     Betweeii  these  two  is 

A  Fig.  38.  B  a  contraction   e,   the    iii'ck.       At    the 

back  of  the  head  is  tlic  surface  of  the 
Joi)it,  by  means  of  wliich  it  fits  on  to1[ 
the  anvil.  Below  the  neck,  where 
the  handle  begins,  project  two  pro- 
cesses, the  long  1,  also  called  joj'o- 
cessns  Folidnus  and  py.  gracilis,  and 
the  short  b,  also  called  ^:>n  hre'vis. 
The  long  process  has  the  proportion- 
ate length  shewn  in  the  figure,  in 
children  only  ;  in  adults  it  appears  to 
l>e  absorl)ed  down  to  a  little  stump. 
It  is  directed  forwards,  and  is  covered 
by  the  bands  which  fasten  the  hammer 
in  front.  The  short  process  b,  on  the  other  hand,  is  directed  towards  the  drumskin, 
and  presses  its  upper  part  a  little  forwards.  From  the  point  of  this  process  b  to 
the  point  of  the  handle   m  the  hammer  is   attached  to  the  upper  portion  of  the  ^ 


Right  haimiifr,  A  fniin  the  fniiit,  li  from  beliiiul.   cp,  liead, 
"  c  neck,  b  siidit,  1  long  process,  ni  handle.        Siirface  of 
the  joint. 


Left  temporal  bone  of  a  newly-born  child,  with  the  audi-  Ri^ht  dnnnskin\viththehaninier,seenfiii>n  the 

tory  ossicles  in  nitii.     Sta,  spina  tynipSnica  anterior.  inside.   Theinnerlayerof  thefoldofnuicons 

Stp,  spina  tympanica  posterior.     Mcp,  head  of  the  menibiune  belonging   t<i  the  haniiner  (see 

hammer.     Mb  short.  Ml  long  process  of  hammer.     ,T  below)  is  removed      Stji.  spiii.-i  tym))rinica 

anvil.-  .S  stirrup.  p„.st.     .Mrp,  head  of  tlie  luiiimi.-r. '  Ml,  long 

process  of  hammer,  ma.liuairifii  turn  mallei 
ant.  1  ehm-diitympani.  ■!  Kuslachiaii  tube. 
*  Tendon  of  the  M.  tensor  tynipaiii,  c\it 
through  close  to  its  insertion. 

drumskin,   in  such  a  manner  that  the  point  of  the  handle  draws  the  drumskin 
considerably  towards  the  inner  part  of  the  ear. 

Fig.  39  above  shews  the  hammer  in  its  natural  position  as  seen  from 
without,  after  the  drumskin  has  been  removed,  and  fig.  40  shews  the  hammer 
lying  against  the  drumskin  as  seen  from  within.     The  hammer  is  fastened  along 


*  [Stapes  is  u.sually  called  stai'peez.     It  is       a  contraction  for  statq)^ 
not  a  classical  word,  and  is  usually  received  as       classical. —  Translator.'] 


foot-rest,  also  not 


132  SYMPATHETICALLY  VIBRATING  PARTS  OF  THE  EAR.        part  k 

the  upper  margin  of  the  drumskiu  by  a  fold  of  mucous  membrane,  within  which 
run  a  series  of  rather  stiff  bundles  of  tendinous  fibres.  These  straps  arise  in  a 
line  which  passes  from  the  processus  Folianus  (fig.  38,  1),  above  the  contraction  of 
the  neck,  towards  the  lower  end  of  the  surface  of  the  joint  for  the  anvil,  and  in 
elderly  people  is  developed  into  a  prominent  ridge  of  bone.  The  tendinous  bands 
or  ligaments  are  strongest  and  stiffest  at  the  fi'ont  and  back  end  of  this  line  of 
insertion.  The  front  portion  of  the  ligament,  lig.  mallei  anterius  (fig.  40,  ma), 
siirrounds  the  processus  Folianus,  and  is  attached  jmrtly  to  a  bony  spine  (figs.  39 
and  40,  Stp)  of  the  osseous  ring  of  the  drum,  which  projects  close  to  the  neck  of 
the  hammer,  and  partly  to  its  under  edge,  and  partly  falls  into  a  bony  fissure 
which  leads  towards  the  articulation  of  the  jaw.  The  back  portion  of  the  same 
ligament,  on  the  other  hand,  is  attached  to  a  sharp-edged  bony  ridge  projecting 
inwards  from  the  drumskin,  and  parallel  to  it,  a  little  above  the  opening,  through 

II  which  a  traversing  nerve,  the  chorda  tympanl  (fig.  40,  1,  1,  p.  131c),  enters  the  bone. 
This  second  bundle  of  fibres  may  be  called  the  lig.  mallei  posterius.  In  fig.  39 
(p.  131c)  the  origin  of  this  ligament  is  seen  as  a  little  projection  of  the  ring  to 
which  the  drumskin  is  attached.  This  projection  bounds  towards  the  right  the 
upper  edge  of  the  opening  for  the  drumskin,  which  begins  to  the  left  of  Stp,  exactly 
at  the  place  where  the  long  process  of  the  anvil  makes  its  appearance  in  the  figure. 
These  two  ligaments,  front  and  back,  taken  together  form  a  moderately  tense 
sinewy  chord,  round  which  the  hammer  can  turn  as  on  an  axis.  Hence  even  when 
the  two  other  ossicles  have  been  carefully  removed,  without  loosening  these  two 
ligaments,  the  hammer  will  remain  in  its  natural  position,  although  not  so  stiffly  as 
before. 

The  middle  fibres  of  the  broad  ligamentous  band  above  mentioned  pass  outwards 
towards  the  upper  bony  edge  of  the  drumskin.  They  are  comparatively  short  and 
are   known   as   lig.   mallei   externiuii.     Arising  above   the   line   of  the   axis  of  the 

^  hanuner,  they  prevent  the  head  from  turning  too  far  inwards,  and  the  handle  with 
the  drumskin  from  turning  too  far  outwards,  and  oppose  any  down-dragging  of  the 
ligament  forming  the  axis.  The  first  effect  is  increased  by  a  ligament  (lig.  mallei 
superius)  which  passes  from  the  processus  Folianus,  upwards,  into  the  small  slit, 
between  the  head  of  the  hammer  and  the  wall  of  the  drum,  as  shewn  in  fig.  40 
(p.  131c). 

It  nuist  be  observed  that  in  the  upper  part  of  the  channel  of  the  Eustachian 
tube,  there  is  a  muscle  for  tightening  the  drumskin  (m.  tensor  tympani),  the  tendon 
of  which  passes  obliquely  across  the  cavity  of  the  drum  and  is  attached  to  the 
upper  part  of  the  handle  of  the  hammer  (at*,  fig.  40,  p.  131c).  This  muscle 
must  be  regarded  as  a  moderately  tense  elastic  band,  and  may  have  its  tension 
temporarily  much  increased  by  active  contraction.  The  effect  of  this  muscle  is 
also  principally  to  draw  the  handle  of  the  hammer  inwards,  together  with  the 
drumskin.     But  since  its  point  of  attachment  is  so  close  to  the  ligamentous  axis„ 

lithe  chief  part  of  its  pull  acts  on  this  axis,  stretching  it  as  it  draws  it  inwards. 
Here  we  must  observe  that  in  the  case  of  a  rectilinear  inextensible  cord,  which 
is  moderately  tense,  such  as  the  ligamentous  axis  of  the  hammer,  a  slight  force 
which  pulls  it  sideways,  suffices  to  produce  a  very  considerable  increase  of  tension. 
This  is  the  case  with  the  present  arrangement  of  stretching  muscles.  It  should 
also  be  remembered  that  quiescent  muscles  not  excited  by  innervation,  are  always 
stretched  elastically  in  the  living  body,  and  act  like  elastic  bands.  This  elastic 
tension  can  of  course  be  considerably  increased  by  the  innervation  which  brings 
the  muscles  into  action,  but  such  tension  is  never  entirely  absent  from  the  majority 
of  our  muscles. 

The  anvil,  which  is  shewn  separately  in  fig.  41,  resembles  a  double  tooth  with 
two  fangs;  the  surface  of  its  joint  with  the  hammer  (at  *,  fig.  41),  replacing  the 
masticating  surface.  Of  the  two  roots  of  the  tooth  which  are  rather  widely 
separated,  the  upper,  directed  backwards,  is  called  the  short  iwocess  b  ;  the  other, 
thinner  and  directed  downwards,  the   long  process  of  the  anvil  1.      At  the  tip  of 


SYMPATHETICALLY  VIBRATING  PARTS  OF  THE  EAR. 


VM) 


Eight  anvil.    A  medial  .suifat 
body,    b  short,  1  loiiji  i)io 
cularis  or  os  orbiculare. 
the  head  of  the  hammer, 
on  the  wall  of  the  drum. 


the  latter  is  the  knob  which  articulates  with  the  stirrup.     The  tip  of  the  sliort 
process,   on  the  other  hand,   by   means   of  a   short  ligament  and  an  imperfectly 

developed  joint  at  its  under  surface,  is  con- 
nected with  the  back  wall  of  the  cavity  of 
the  dnnn,  at  the  spot  where  this  passes 
backwards  into  the  air  cavities  of  the  mastoid 
process  behind  the  eax*.  The  joint  between 
anvil  and  hanuner  is  a  curved  depression  of 
a  rather  irregular  form,  like  a  saddle.  In 
its  action  it  may  be  compared  with  the  joints 
of  the  well-known  Breguet  watchkcys,  which 
have  rows  of  interlocking  teeth,  offering 
scarcely  any  resistance  to  revolution  in  one 
direction,  but  allowing  no  revolution  what-  ^ 
\rti'uEn^wiu;  ever  in  the  other.  Interlocking  teeth  of 
Surface  resting  ^yjjg  i^j^^^^  ^rc  developed  upou  the  under  side 
of  the  joint  between  hammer  and  anvil. 
The  tooth  on  the  hanuner  projects  towards  the  drumskin,  that  of  the  anvil  lies 
inwards ;  and,  conversely,  towards  the  upper  end  of  the  hollow  of  the  joint,  the 
anvil  projects  outwards,  and  the  hammer  inwards.  The  consequence  of  this 
arrangement  is  that  when  the  hammer  is  drawn  inwards  by  the  handle,  it  bites 
the  anvil  firmly  and  carries  it  with  it.  Conversely,  when  the  drumskin,  with  the 
hammer,  is  driven  outwards,  the  anvil  is  not  obliged  to  follow  it.  The  interlocking 
teeth  of  the  surfaces  of  the  joint  then  separate,  and  the  surfaces  glide  over  each 
other  with  very  little  friction.  This  arrangement  has  the  very  great  advantage  of 
preventing  any  possibility  of  the  stirrup's  being  torn  away  from  the  oval  window, 
when  the  air  in  the  auditory  passage  is  considerably  rarefied.  There  is  also  no 
danger  from  driving  in  the  hammer,  as  might  happen  when  the  air  in  the  auditory  ^ 
passage  was  condensed,  because  it  is  powerfully  opposed  by  the  tension  of  the 
drumskin,  which  is  drawn  in  like  a  funnel. 

When  air  is  forced  into  the  cavity  of  the  drum  in  the  act  of  swallowing,  the 
contact  of  hammer  and  anvil  is  loosened.  Weak  tones  in  the  middle  and  upper 
regions  of  the  scale  are  then  not  heard  much  more  weakly  than  usual,  but  stronger 
tones  are  very  sensibly  damped.  This  may  perhaps  be  explained  by  supposing  that 
the  adhesion  of  the  articulating  surfaces  suflices  to  transfer  weak  motions  from  one 
bone  to  the  other,  but  that  strong  impulses  cause  the  siu-faces  to  slide  over  one 
another,  and  hence  the  tones  due  to  such  impulses  must  be  enfeebled. 

Deep  tones  are  damped  in  this  case,  whether  they  are  strong  or  weak,  ^jerhaps 
because  these  always  require  larger  motions  to  become  audible.* 

Another  important  eftect  on  the  apprehension  of  tone,  which  is  due  to  the  above 
arrangement  in  the  articulation  of  hammer  and  anvil,  will  have  to  be  considered  in 
relation  to  combinational  tones.     [See  p.  158/a]  ^ 

Since  the  attachment  of  the  tip  of  the  short  process  of  the  anvil  lies  sensibly 
inwards  and  above  the  ligamentous  axis  of  the  liaunner,  the  head  of  the  hammer 
separates  from  the  articulating  surface  between  hammer  and  anvil,  when  the  head 
is  driven  outwards,  and  therefore  the  handle  and  drumskin  are  driven  inwards. 
The  consequence  is  that  the  ligaments  holding  the  anvil  against  the  hammer,  and 
on  the  tip  of  the  short  process  of  the  anvil,  are  sensibly  stretched,  and  hence  the 
tip  is  raised  from  its  osseous  support.  Consequently  in  the  normal  position  of  the 
ossicles  for  hearing,  the  anvil  has  no  contact  with  any  other  bone  but  the  hammer, 
and  both  bones  are  held  in  position  only  by  stretched  ligaments,  which  are  tolerably 
tight,  so  that  only  the  revolution  of  the  hammer  about  its  ligamentous  axis  remains 
comparatively  free. 

The  third  ossicle,  the  stirnq^,  shewn  separately  in  fig.   42,   has  really  a  most 
striking  resemblance  to  the  implement  after  which  it  has  been  named.     The  foot   P> 
*  On  this  point  see  Part  II.  Chapter  IX. 


134  SYMPATHETICALLY  VIBltATING  I'AHTS  OF  THE  EAR.        part  i. 

is  fastened  into  the  membrane  of  the  oval  window,  and  fills  it  all  np,  with  the 
exception  of  a  narrow  margin.  The  head  cp,  has  an  articulating  hole  for  the  tip 
of  the  long  process  of  the  anvil 
(processus  lenticularis,  or  os 
orbiculare).  The  joint  is  sur- 
rounded by  a  lax  membrane. 
When  the  drumskin  is  normally 
drawn  inwards,  the  anvil  presses 
on  the  stirrui),  so  that  no  tighter  — 

ligamentous      fastening      of      tlie      j^.^j^^  ^^j^.^.,,^^ .  ^^^,,  ^  j^.,„^^  ^^.j^j^i^  ^  j^,,,,,,  f,,„„t_  ^  fj.^„„  ,3^. 

ioint      is      necessarv.  Every      in-  hind-    B  foot,    cp,  head  or  capituhim.    a  Front,  p  back 

crease  in  the  [)usii  on  tlie  hammer 

arising  from  the  drumskin  also  occasions  an  increase  in  the  push  of  the  stirrup 

^  against  the  oval  window ;  but  in  this  action  the  upper  and  somewhat  looser 
margin  of  its  foot  is  more  displaced  than  the  under,  so  that  the  head  rises  slightly ; 
this  motion  again  causes  a  slight  elevation  of  the  tip  of  the  long  process  in  the 
anvil,  in  the  direction  conditioned  by  its  position,  inwards  and  underneath  the 
ligamentous  axis  of  the  hannner. 

The  excursions  of  the  foot  of  the  stirrup  are  always  very  small,  and  according 
to  my  measurements  *  never  exceed  one-tenth  of  a  millimetre  (•00394  or  about 
_i_  of  an  inch).  But  tlie  hammer  when  freed  from  anvil  and  stirrup,  with  its 
handle  moving  outwards,  and  sliding  over  the  articulating  surface  of  the  anvil,  can 
make  excursions  at  least  nine  times  as  great  as  it  can  execute  when  acting  in 
connection  with  anvil  and  stirrup. 

The  first  advantage  of  the  apparatus  belonging  to  the  drum  of  the  ear,  is  that 
the  whole  sonorous  motion  of  the  comparatively  wide  surface  of  the  drumskin  (ver- 
tical diameter  9  to  10  millimetres  [or  0-35  to  0-39  inches],  just  over  one-third  of  an 

II  inch;  horizontal  diameter,  7-5  to  9  millimetres  [or  0-295  to  0-35  inches],  that  is 
about  five-sixths  of  the  former  dimensions)  is  collected  and  transferred  by  the 
ossicles  to  the  relatively  much  smaller  surface  of  the  oval  window  or  of  the  foot  of 
the  stirrup,  which  is  only  1-5  to  3  millimetres  [0-06  to  0'12  inches]  in  diameter. 
The  surface  of  the  drumskin  is  hence  15  to  20  times  larger  than  that  of  the  oval 
window. 

In  this  transference  of  the  vibrations  of  air  into  the  labyrinth  it  is  to  be  observed 
that  though  the  particles  of  air  themselves  have  a  comparatively  large  amplitude  of 
vibration,  yet  their  density  is  so  small  that  they  have  no  very  great  moment  of  inertia^ 
and  consequently  when  their  motion  is  impeded  by  the  drumskin  of  the  ear,  they 
are  not  capable  of  presenting  much  resistance  to  such  an  impediment,  or  of  exert- 
ing any  sensible  pressure  against  it.  The  fluid  in  the  labyrinth,  on  the  other  hand, 
is  much  denser  and  heavier  than  the  air  in  the  auditory  passage,  and  for  moving  it 
rapidly  backwards  and  forwards  as  in  sonorous  oscillations,  a  far  greater  exertion  of 

^pressure  is  required  than  was  necessary  for  the  air  in  the  auditory  passage.  On 
the  other  hand  the  amplitude  of  the  vibrations  performed  by  the  fluid  in  the  laby- 
rinth are  relatively  very  small,  and  extremely  minute  vibrations  will  in  this  case 
siiffice  to  give  a  vibratory  motion  to  the  terminations  and  appendages  of  the  nerves^ 
which  lie  on  the  very  limits  of  microscopic  vision. 

The  mechanical  problem  which  the  apparatus  Avithin  the  drum  of  the  ear  had 
to  solve,  was  to  transform  a  motion  of  great  amplitude  and  little  force,  such  as  im- 
pinges on  the  drumskin,  into  a  motion  of  small  amplitude  and  great  force,  such  as 
had  to  be  communicated  to  the  fluid  in  the  labyrinth. 

A  problem  of  this  sort  can  be  solved  by  various  kinds  of  mechanical  apparatus, 
such  as  levers,  trains  of  pulleys,  cranes,  and  the  like.  The  mode  in  which  it  is 
solved  by  the  apparatus  in  the  drum  of  the  ear,  is  quite  unusual,  and  very  peculiar. 

*  Helmholtz,  '  Llechanism  of  the  Auditory  attempt  is  made  to  prove  the  correctness  of 
Ossicles,'  in  Pflueger's  Archir  fur  ihysio-  the  account  of  this  mechanism  given  in  the 
logic,   voL    i.    pp.   34-43.      In    this   paper   an       text. 


CHAP.  VI.    SYMPATHETICALLY  VIBRATING  PARTS  OF  THE  EAR.  13.5 

A  leverage  is  certainly  employed,  but  only  to  a  moderate  extent.  Tlie  tip  of 
the  handle  of  the  hammer,  on  which  the  pidl  of  the  drumskin  first  acts,  is  about 
once  and  a  half  as  far  from  the  axis  of  rotation  as  that  point  of  the  anvil  which 
presses  on  the  stirrup  (see  fig.  39,  p.  131r).  The  handle  of  the  hammer  consequently 
forms  the  longer  arm  of  a  lever,  and  the  pressure  on  the  stirrup  will  be  once  and  a 
half  as  great  as  that  which  drives  in  the  hammer. 

The  chief  means  of  reinforcement  is  due  to  the  form  of  the  drumskin.  It  has 
been  already  mentioned  that  its  middle  or  nnvd  (umbilicus)  is  drawn  inwards  by 
the  handle,  so  as  to  present  a  funnel  shape.  Pnit  the  meridian  lines  of  this  funnel 
drawn  from  the  navel  to  the  circumference,  are  not  straight  lines ;  they  are  slightly 
convex  on  the  outer  side.  A  diminution  of  pressure  in  the  auditory  passage  in- 
creases this  convexity,  and  an  augmentation  diminishes  it.  Now  the  tension  caused 
in  an  inextensible  thread,  having  the  form  of  a.  Hat  arch,  by  a  force  acting  perpen- 
dicular to  its  convexity,  is  very  considerable.  It  is  well  known  that  a  sensible  force  ^ 
must  be  exerted  to  stretch  a  long  thin  string  into  even  a  tolerably  straight  horizon- 
tal line.  The  force  is  indeed  very  much  greater  than  the  weight  of  the  string  which 
pulls  the  string  from  the  horizontal  position. '■■  In  the  case  of  the  drumskin,  it  is 
not  gravity  which  prevents  its  radial  fibres  from  straightening  themselves,  but  partly 
the  pressure  of  the  air,  and  partly  the  elastic  pull  of  the  circular  fibres  of  the  mem- 
brane. The  latter  tend  to  contract  towards  the  axis  of  the  funnel-shaped  mem- 
brane, and  hence  produce  the  inflection  of  the  radial  fibres  towards  this  axis.  By 
means  of  the  variable  pressure  of  air  during  the  sonorous  vibrations  of  the  at- 
mosphere this  pull  exerted  by  the  circular  fibres  is  alternately  strengthened  and 
weakened,  and  produces  an  effect  on  the  point  where  the  radial  fibres  are  attached 
to  the  tip  of  the  handle  of  the  hammer,  similar  to  that  which  would  happen  if  we 
could  alternately  increase  and  diminish  the  weight  of  a  string  stretched  horizontally, 
for  this  would  produce  a  proportionate  increase  and  decrease  in  the  pull  exerted  by 
the  hand  which  stretched  it. 

In  a  horizontally  stretched  string  such  as  has  been  just  described,  it  shoiild  be 
further  remarked  that  an  extremely  small  relaxation  of  the  hand  is  followed  by  a 
considerable  fall  in  the  middle  of  the  string.  The  relaxation  of  the  hand,  namely, 
takes  place  in  the  direction  of  the  chord  of  the  arc,  and  easy  geometrical  con- 
siderations  show  that  chords  of  arcs  of  the  same  length  and  different,  but  always: 
very  small  curvature,  differ  very  slightly  indeed  from  each  other  and  from  the 
lengths  of  the  arcs  themselves.t  This  is  also  the  case  with  the  drumskin.  An  ex- 
tremely little  yielding  in  the  handle  of  the  hammer  admits  of  a  very  considerable 
change  in  the  x;urvature  of  the  drumskin.  The  consequence  is  that,  in  sonorous 
vibrations,  the  parts  of  the  drumskin  which  lie  between  the  inner  attachment  of 
this  membrane  to  the  hammer  and  its  outer  attachment  to  the  ring  of  the  drum, 
are  able  to  follow  the  oscillations  of  the  air  with  considerable  freedom,  while  the 
motion  of  the  air  is  transmitted  to  the  handle  of  the  hammer  with  much  diminished 
amplitude  but  much  increased  force.  After  this,  as  the  motion  passes  from  the  U 
handle  of  the  hammer  to  the  stirrup,  the  leverage  already  mentioned  causes 
a  second  and  more  moderate  reduction  of  the  amplitude  of  vibration  with  corre- 
sponding increase  of  force. 

We  now  proceed  to  describe  the  innermost  division  of  the  organ  of  hearing, 
called  the  lahyrinth.  Fig.  43  (p.  134r)  represents  a  cast  of  its  cavity,  as  seen  from 
different  positions.  Its  middle  portion,  containing  the  oval  ivindow  Fv  (fenestra 
vestibulT)  that  receives  the  foot  of  the  stirrup,  is  called  the  vestibule  of  the  lahyrinth, 

*  [The   following   quatrain,    said  to   have  Into  a  horizontal  line, 

been  unconsciously  produced  by  Vince,  as  a  So  as  to  make  it  truly  straight.  — T/wjis^ft^or.] 

corollary   to   one  of   the   propositions   in  his  ,  ^j^^  amount  of   difference  varies  as  the 

'  Mechanics,     will  serve  to   impress   the  fact       ^  ^^^^.^  ^^  ^^^  ^     .  j^  ^f  ^^^  ^^^      If  tl,c  length 
on  a  non-mathematical  reader  :—  ^\  ^j^^  ^^^  ^^^  ^^  ^^^^  ^i^e  distance  of  its  middle 

from  the  chord  be  s,  the  chord  is  shorter  than 


Hence  no  force,  however  great, 

Can  stretch  a  cord,  however  fine,  tl^e  arc  by  the  length  - 


■6     I 


136 


LABYRINTH  OF  THE  EAR. 


From  the  vestibule  proceeds  forwards  and  iinderwards,  a  spiral  canal,  the  snail- 
shell  or  cochlea,  at  the  entrance  to  which  lies  the  round  window  Fc  (fenestra 
cochleae),  which  is  turned  towards  the  cavity  of  the  drum.  Upwards  and  back- 
wards, on  the  other  hand,  proceed  three  semicircular  canals  from  the  vestibule,  the 
ho7-izontal,  front  vertical  and  hack  vertical  semicircular  canals,  each  of  which 
debouches  with  both  its  mouths  in  the  vestibule,  and  each  of  which  has  at  one 
end  a  bottle-shaped  enlargement,  or  ampulla  (ha,  vaa,  vpa).  The  aquaeductus 
vestibrdi  shown  in  the  figure,  Av,  appears  (from  Dr.  Fr,  E.  Weber's  investigations) 
to  form  a  communication  between  the  water  of  the  labyrinth,  and  the  spaces  for 
lymph  within  the  cranium.  The  rough  places  Tsf  and*  are  casts  of  canals  which 
introduce  nerves. 

The  whole  of  this  cavity  of  the  labyrinth  is  filled  with  fluid,  and  surrounded  by 
the  extremely  hard  close  mass  of  the  petrous  bone,  so  that  there  are  only  two 
H  yielding  spots  on  its  walls,  the  two  windows,  the  oval  Fv,  and  the  round  Fc.  Into 
the  first,  as  already  described,  is  fastened  the  foot  of  the  stirrup,  by  a  narrow 
membranous  margin.  The  second  is  closed  by  a  membrane.  When  the  stirrup 
is  driven  against  the  oval  window,  the  whole  mass  of  fluid  in  the  labyrinth  is 
necessarily  driven  against  the  round  window,  as  the  only  spot  where  its  walls  can 
give  way.  If,  as  Politzer  did,  we  put  a  finely  drawn  glass  tube  as  a  manometer 
into  the  round  window,  without  in  other  respects  injuring  the  labyrinth,  the  water 
in  this  tube  will  be  driven  upwards  as  soon  as  a  strong  pressure  of  air  acts  on  the 


Ik  '■■■ 


Re 
vaa 


vpa 


It  In 


A,  left  labyiinth  from  without.  15,  liul 
cochleae  or  voui:  1  window.  Fv,  fen 
sphaerlcus.  li  liorizoutal  st'iniiircu 
semicircular  t-iuuil.  \\ni,  aiiiimllu  of  tlie  liack  vertical  seinii 
semicircular  canals.  .\v,  cast  of  tlie  afiuaeductus  vest'ihul 
little  canals  which  debouch  on  the  pyramis  vestlbuli. 


yrinth  from  withii 
vestibnli,  or  oval 
nal.     ha,  ampulla 


C,  left  labyrinth  from  above.  Fc,  fenestra 
.vindow.  Re,  recessus  elliptlcus.  Rs,  recessus 
if  the  same,  vaa,  ampulla  of  the  front  vertical 
lular  canal,  vc,  common  limb  of  the  two  vertical 
Tsf,  tractus  spiralis  foramlnosus.    "Cast  of  the 


outside  of  the  drumskin  and  causes  the  foot  of  the  stirrup  to  be  driven  into  the  oval 
window. 

The  terminations  of  the  auditory  nerve  are  spread  over  fine  membranous 
^  formations,  which  lie  partly  floating  and  partly  expanded  in  the  hollow  of  the  bony 
labyrinth,  and  taken  together  compose  the  metnlranous  labyrinth.  This  last  has 
on  the  whole  the  same  shape  as  the  bony  labyrinth.  But  its  canals  and  cavities 
are  smaller,  and  its  interior  is  divided  into  two  separate  sections ;  first  the 
titrlcillus  with  the  semicircular  canals,  and  second  the  saccidus  with  the  niem- 
branotis  cochlea.  Both  the  utriculus  and  the  sacculus  lie  in  the  vestibule  of  the 
bony  labyrinth  :  the  utriculus  opposite  to  the  recessus  elliptlcus  (Re,  fig.  43  above), 
the  sacculus  opposite  to  the  recessus  sphaerhtis  (Rs).  These  are  floating  bags 
filled  with  water,  and  only  touch  the  wall  of  the  labyrinth  at  the  point  where  the 
nerves  enter  them. 

The  form  of  the  utriculus  with  its  membranous  semicircular  canals  is  shewn  in 
fig.  44.  The  ampullae  project  much  more  in  the  membranous  than  in  the  bony  semi- 
circular canals.  According  to  the  recent  investigations  of  Riidinger,  the  mem- 
branous semicircular  canals  do  not  float  in  the  bony  ones,  but  are  fastened  to  the 
convex  side  of  the  latter.     In  each  ampulla  there  is  a  pad-like  prominence  directed 


CHAP 


COCHLEA  OF  THE  EAR. 


13' 


Utriculus  and  nieinbiunous  semicir- 
cular canals  (left  side)  seen  from 
without,  va front,  vp  back  vertical, 
h  horizontal  semicircular  canal. 


inwards,   into   which  tibrilos  of    the  auditory  nerve  enter;    and   on    tlic   utricuhis 

there  is  a  phxce  whicli   is  flatter  and  thickened.     The  peculiar  manner  in   which 

the  nerves  terminate  in  this  place  will   be  de.scriV)ed 
Fig.  44.  ^ 

^.^  liereafter.      Whether  these,  and   the  whole  apparatus 

of  the  semicircular  canals,  assist  in  the  sensation  of 
hearing,  has  latterlv  been  rendered  very  douV)tful.  [See 
p.  1516.] 

In  the  inside  of  the  utriculus  is  found  the  auditory 
sand,  consisting  of  little  crystals  of  lime  connected  by 
means  of  a  nuicous  mass  with  each  other  and  witli  the 
thickened  places  where  the  nerves  are  so  abundant. 
In  the  hollow  of  the  bony  vestibide,  near  the  utriculus, 
and  fastened  to  it,  but  not  communicating  with  it,  lies 
the  sacculus,  also  provided  with  a  similar  thickened  H 
spot  full  of  nerves.  A  narrow  canal  connects  it  with 
the  canal  of  the  membranous  cochlea.  As  to  the  cavity 
of  the  cochlea,  we  see  by  fig.  43  opposite,  that  it  is 
exactly  similar  to  the  shell  of  a  garden  snail  ;  but  the  canal  of  the  cochlea  is 
divided  into  two  almost  completely  separated  galleries,  by  a  transverse  partition, 
partly  bony  and  partly  membranous.  These  galleries  communicate  only  at  the 
vertex  of  the  cochlea  through  a  small  opening,  the  hHlcotrema,  bounded  by  the 
h'lmahis  or  hook-shaped  termination  of  its  central  axis  or  tnodi'olus.  Of  the  two 
galleries  into  which  the  cavity  of  the  bony  cochlea  is  divided,  one  communicates 
directly  with  the  vestibule  and  is  hence  called  the  vestibide  gallery  (sciila  vestiball). 
The  other  gallery  is  cut  off  from  the  vestibule  by  the  membranous  partition,  but 
just  at  its  base,  where  it  begins,  is  the  round  window,  and  the  yielding  membrane, 
which  closes  this,  allows  the  fluid  in  the  gallery  to  exchange  vibrations  with  the 
air  in  the  drum.     Hence  this  is  called  the  drum  galleri/  (scala  tymp;ail).  ^ 

Finally,  it  must  be  observed  that  the  membranous  partition  of  the  cochlea  is 
not  a  single  membrane,  but  a  membranous  canal  (ductus  cochlearis).     Its  inner 
Pj^  ^g  margin  is  turned  tovvards  the  central  axis  or 

modiolus,  and  attached  to  the  rudimentary 
bony  partition  (lamina  spiralis).  A  part  of 
the  opposite  external  surface  is  attached  to  the 
inner  surface  of  the  bony  gallery.  Fig.  45 
shews  the  bony  parts  of  a  cochlea  which  has 
been  laid  open,  and  fig.  46  (p.  138«),  a  trans- 
verse section  of  the  canal  (which  is  imperfect 
on  the  left  hand  at  bottom).  In  both  figures 
Ls  denotes  the  bony  part  of  the  partition,  and 
in  fig.  46  V  and  b  are  the  two  unattached  parts 
of  the  membranous  canal.  The  transverse  H 
section  of  this  canal  is,  as  the  figure  shews, 
nearly  triangular,  so  that  an  angle  of  the 
triangle  near  Lis  is  attached  to  the  edge  of 
the  bony  partition.  The  conmiencement  of 
the  ductus  cochlearis  at  the  base  of  the 
cochlea,  communicates,  as  already  stated,  by 
means  of  a  narrow  membranous  canal  with 
the  sacculus  in  the  vestil)ule.  Of  the  two  un- 
attached strips  of  its  membranous  walls,  the 
one  facing  the  vestibule  gallery  is  a  soft  mem- 
brane, incapable  of  ottering  nnich  YCiAai-AWce^Reissner's  membrane  (membrana  vesti- 
bularis, V,  fig.  46,  p.  138a);  but  the  other,  the  membrana  basilclris  (b),  is  a  firm, 
tightly  stretched,  elastic  membrane,  striped  radially,  corresponding  to  its  radial 
fibres.     It  splits  easily  in  the  direction  of  these  fibres,  shewing  that  it  is  but  loosely 


Fee 

Bony  cochlea  (right  side)  opened  in  fiont. 
modiolus.  I.s,  lainliia  siiiialis.  II,  lifn 
Fee, fenestra  cochleae.  ^St'ctionof  tliejiai 
of  the  cochlea.    1  iL'i)iiev  extremity  of  the 


138 


COCHLEA  OF  THE  EAR. 


fransverse  section  of  a  spire  of  a  cochlea  whichhas  been, 
softened  in  hydrochloric  acid.  Ls,  lamina  spiralis.  Lis,, 
limbus  laminae  spiralis.  Sv,  scala  ve.stlbftll.  St,  scala 
tymp5nl.  Dc,  ductus  cochlearis.  Lsp,  Itgamentum 
spirale.  v,  membrana  vestibularis,  h,  membrana  Mst- 
laris.  e,  outer  wall  of  the  ductus  cochlearis._  *  its  fillet. 
The  dotted  lines  shew  sections  of  the  membrana  tectoria 
and  the  auditory  roils. 


comit'cted  in  a  direction  tninsverse  to  them.  The  terminations  of  the  nerves  of 
the  cochlea  and  their  appendages,  are  attached  to  the  membrana  basilaris,  as  is. 
shewn  by  the  dotted  lines  in  fig.  46. 

When  the  drnmskin  is  driven  inwards  by  increased  pressure  of  air  in  the  auditory 
passage,  it  also  forces  the  auditory  ossicles  inwards,  as  already  explained,  and  as  a. 
consequence  the  foot  of  the  stiiTup 
penetrates  deeper  into  the  oval  window. 
Tlie  fiuid  of  the  labyrinth,  being  sur- 
rounded in  all  other  places  by  firm 
bony  walls,  has  only  one  means  of 
escape, — the  round  window  with  its 
3'ielding  membrane.  To  reach  it,  the 
fluid  of  the  labyrinth  must  either  pass 
^through  the  helicotrema,  the  narrow 
opening  at  the  vertex  of  the  cochlea, 
flowing  over  from  the  vestibule  gallery 
into  the  drum  gallery,  or,  as  it  would 
pr(jl>al)ly  not  have  sufficient  time  to  do 
this  in  the  case  of  sonorous  vibrations, 
press  the  membranous  partition  of  the 
cochlea  against  the  drum  gallery.  The 
converse  action  must  take  place  when 
the  air  in  the  auditory  passage  is  rare- 
fied. 

Hence  the  sonorous  vibrations   of  the   air  in   the   outer  auditory  passage  are 
finally  transferred  to  the  membranes  of  the  labyrinth,  more  especially  those  of  the 
cochlea,  and  to  the  expansions  of  the  nerves  upon  them. 
^        The  terminal  expansions  of  these  nerves,  as  I  have  already  mentioned,  are  con- 
nected with  very  small   elastic  appendages,   which  appear  adapted   to   excite  the 

nerves  by  their  vibrations. 

The  nerves  of  the  vestibule  terminate  in  the  thickened  places  of  the  bags  of 

the    membranoxis    labyrinth,   already    mentioned  p,o  47 

(p.   137rt),  where  the  tissue   has  a  greater  and 

almost  cartilaginous  consistency.     One  of   these 

places,  provided  with  nerves,  projects  like  a  fillet 

into  the  inner  part  of  the  ampulla  of  each  semi- 
circular canal,  and  another  lies  on  each  of  the  little 

bags  in  the  vestibule.     The  nerve  fibres  here  enter 

between  the  soft  cylindrical  cells  of  the  fine  cuticle 

(("•pithrlTum)  which  covers  the  internal  surface  of 

the  fillets.      Projecting  from  the  internal  surface 
^of  this  epithelium  in  the  ampullae,  Max  Schultze 

discovered  a  number  of  very  peculiar,  stiff,  elastic 

hairs,  shewn  in  fig.  47.     They  are  much  longer 

than  the  vibratory  hairs  of  the  ciliated  epithe'lium 

(their  length   is   ttV   of  a   Paris  line  [or  -00355 

English  inch]  in  the  ray  fish),  brittle,  and  running 

to  a  very  fine  point.     It  is  clear  that  fine  stiff 

hairs  of  this  kind  are  extremely  well  adapted  for 

moving  sympathetically  with  the  motion  of  the 

fluid,   and   hence  for  producing  mechanical  irri- 
tation in  the  nei've  fibres  which  lie  in  the  soft 

epithelium  between  their  roots. 

According  to  Max  Schultze,  the  corresponding 

thickened  fillets  in  the  vcstibides,  where  the  nerves  terminate,  have  a  similar  soft 

epithelium,   and    have    short    hairs  which    are    easily   destroyed.      Close   to    these 


COCHLEA  OF  THK  EAR. 


139 


surfaces  which  are  covered  with  nerves,  lie  the  calcareous  concretions,  called 
auditory  stones  (otoliths),  which  in  fishes  form  connected  convexo-concave  solids, 
shewing  on  their  convex  side  an  impression  of  the  nerve  fillet.  In  human  beings, 
on  the  other  hand,  the  otoliths  are  heaps  of  little  crystalline  bodies,  of  a  longish 
angular  form,  lying  close  to  the  membrane  of  the  little  bags,  and  apparently 
attached  to  it.  "  These  otoliths  seem  also  extremely  well  suited  for  producing  a 
mechanical  irritation  of  the  nerves  whenever  the  fluid  in  the  labyrinth  is  suddenly 
auitated.  The  fine  light  membrane,  with  its  interwoven  nerves,  probably  instantly 
follows  the  motion  of  the  fluid,  whereas  the  heavier  crystals  are  set  more  slowly  in 
motion,  and  hence  also  yield  up  their  motion  more  slowly,  and  thus  partly  drag 
and  partly  squeeze  the  adjacent  nerves.  This  would  satisfy  the  same  conditions 
of  exciting  nerves,  as  Heideuhain's  tetdnonwtor.  By  this  instrument  the  nerve 
which  acts  on  a  muscle  is  exposed  to  the  action  of  a  very  rapidly  oscillating  ivory 
hammei-,  which  at  every  blow  squeezes  without  bruising  the  nerve.  A  powerful 
and  continuous  excitement  of  the  nerve  is  thus  produced,  which  is  shewn  by  a^l 
powerful  and  continuous  contraction  of  the  corresponding  muscle.  The  above  parts 
of  the  ear  seem  to  be  well  suited  to  produce  similar  mechanical  excitement. 

The  construction  of  the  cochlea  is  much  more  complex.  The  nerve  fibres  enter 
through  the  axis  or  modiolus  of  the  cochlea  into  the  bony  part  of  the  partition, 
and  then  come  on  to  the  membranous  part.  Where  they  reach  this,  peculiar 
formations  were  discovered  quite  recently  (1851)  by  the  Marchese  Corti,  and  have 
been  named  after  him.     On  these  the  nerves  terminate. 

The  expansion  of  the  cochlean  nerve  is  shewn  in  fig.  48.  It  enters  through 
the  axis  (2)  and  sends  out  its  fibres  in  a  radial  direction  from  the  axis  through  the 

bony  partition  (1,  3,  4), 

as  far   as    its    margins. 

4  At  this  point  the  nerves 

pass  under  the  com- 
mencement of  the  mem-  ^ 
brana  basiliiris,  pene- 
trate this  in  a  series  of 
openings,  and  thus  reach 
the  ductus  cochlearis 
and  those  nervous, 
elastic  formations  which 
lie  on  the  inner  zone 
(Zi)  of  the  membrane. 

The  margin   of   the 
bony  partition  (a  to  b, 
fig.  49,  p.  140a),  and  the 
-^  _     -— -  inner  zone  of  the  mem- 

brana  basilaris  (a  a')  arc  shewn  after  Hensen.  The  under  side  of  the  figure* 
corresponds  with  the  scala  tympani,  the  upper  with  the  ductus  cochlearis.  Here  h  11 
at  the  top  and  k  at  the  bottom,  are  the  two  plates  of  the  bony  partition,  between 
which  the  expansion  of  the  nerve  b  proceeds.  The  upper  side  of  the  bony  parti- 
tion bears  a  fillet  of  close  ligamentous  tissue  (Z,  fig.  49,  also  shewn  at  Lis,  fig.  46, 
p.  138o),  which,  on  account  of  the  ,toothlike  impressions  on  its  upper  side,  is  called 
the  toothed  layer  {zona  denticulata),  and  which  carries  a  peculiar  elastic  pierced 
membrane,  Corti's  membrane,  M.C.  fig.  49.  This  membrane  is  stretched  parallel 
to  the  mem] )rana  basilaris  as  far. as  the  bony  wall  on  the  outer  side  of  the  duct, 
and  is  there  attached  a  little  above  the  other.  Between  these  two  membranes 
lie  the  parts  in  and  on  which  the  nerve  fibres  terminate. 

Among  these  Corti's  arches  (over  g  in  fig.  49)  are  relatively  the  most  solid 
formations.     The  series  of  these  contiguous   arches   consists  of  two  series  of  rods 


Lsp 


^'  [  As  the  engraving  would  have  been  too 
wide  for  the  page  if  placed  in  its  proper  hori- 
zontal position,  it  has  been  printed  vertically  ; 


the  Irft  side  consequently  corresponds  to  the 
vpprr,  and  its  riiiht  to  the  under  side. — Trans- 
lator A 


140 


COCHLEA  OF  THE  EAR. 


•i 


%i 


ov  ahres,  an  external  and  an  internal.  A  single  pair  of  these  is  shewn  in  fig.  .50, 
A,  below,  and  a  short  series  of  them  in  fig.  50,  B,  attached  to  the  membrana 
basilaris,  and  at  +  also  connected  with  the  pierced  P^^  ^^, 

tissue,  into  which  fit  the  terminal  cells  of  the  nerves 
(fig.  49,  c),  which  will  be  more  fully  described  pre- 
sently. These  formations  are  shewn  in  fig.  51, 
(p.  14yj,  c),  as  seen  from  the  vestibule  gallery;  a  is 
the  denticulated  layer,  c  the  openings  for  the 
nerves  on  the  internal  margin  of  the  membrana 
basilaris,  its  external  margin  being  visible  at  u  u ; 
d  is  the  inner  series  of  Corti's  rods,  e  the  outer ; 
over  these,  between  e  and  x  is  seen  the  pierced 
membrane,  against  whicli  lie  the  terminal  cells  of 

51  the  nerves. 

The  fibres  of  the  first,  or  outer  series,  are  flat, 
somewhat  S-shaped  formations,  having  a  swelling 
at  the  spot  where  they  rise  from  the  membrane  to 
which  they  are  attached,  and  ending  in  a  kind  of 
articulation  which  serves  to  connect  them  with  the 
second  or  inner  series.  In  fig.  51,  p.  141,  at  d 
will  be  seen  a  gi-eat  number  of  these  ascending 
fiV)res,  lying  beside  each  other  in  regular  succession. 
In  the  same  way  they  may  be  seen  all  along  the 
membrane  of  the  cochlea,  close  together,  so  that 
there  must  be  many  thousands  of  them.  Their 
sides  lie  close  together,  and  even  seem  to  be  con- 
nected, leaving  however  occasional  gaps  in  the  line 

^(if  connection,  and  the.se  gaps  are  probably  tra- 
versed by  nerve  fibres.  Hence  the  fibres  of  the 
first  series  as  a  whole  form  a  stitl*  layer,  which 
endeavours  to  erect  itself  when  the  natural  fasten- 
ings no  longer  resist,  but  allows  the  membrane  on 
which  they  stand  to  crumple  up  between  the  at- 
tachments d  and  e  of  Corti's  arches. 

The  fibers  of  the  second  or  inner  Sf^rir's,  which 
form  the  descending  part  of  the  arch  e,  fig.  50, 
below,  are  smooth,  flexible,  cylindrical  threads 
with  thickened  ends.  The  upper  extremity  forms 
a  kind  of  joint  to  connect  them  with  the  fibres 
of  the  first  series,  the  lower  extremity  is  enlarged  in  a  bell  shape  and  is  attached 
closely  to  the  membrane  at  the  base.     In  the  microscopic  preparations  they  gene- 

51  Fig.  50. 

A  B 


A,  external  ami  internal  rod  in  connection  seen  in  protile.  B,  membrana  basilaris  (b)  with  the 
terminal  fascTculI  of  nerves  (n),  and  tlie  internal  and  external  I'ods  (i  and  e)._  1  internal, 
2  external  cells  of  the  floor,  4  attachments  of  the  cells  of  the  cover.     *  '  epitiielium. 

rally  appear  bent  in  various  ways  :  but  there   can  be  no  doubt  that  in  their  natural 
condition  they  are  stretched  with  some  degree  of  tension,  so  that  they  pull  down 


COCHLEA  OF  THE  EAR. 


141 


the  upper  jointed  ends  of  the  fibres  of  the  first  series.  Tlte  fibres  of  the  first 
series  arise  from  the  inner  margin  of  the  membrane,  which  can  be  rehitively  little 
ao-itated,  bnt  the  fibres  of  the  second  series  are  attached  nearly  in  the  middle  of  the 
membrane,  and  this  is  precisely  the  place  where  its  vibrations  will  have  the  greatest 
excursions.  When  the  pressure  of  the  fluid  in  the  drum  gallery  of  the  labyrinth 
is  increased  by  driving  the  foot  of  the  stirrup  against  the  oval  window,  the  mem- 
brane at  the  base  of  the  arches  will  sink  downwards,  the  fibres  of  the  second  series 
be  more  tightly  sti-etched,  and  perhaps  the  corresponding  places  of  the  fibres  of  tlie 
fii-st  series  be  bent  a  little  downwards.  It  does  not,  however,  seem  probable  that 
the  fibres  of  the  first  series  themselves  move  to  any  great  extent,  for  their  lateral 
connections  are  strong  enough  to  make  them  hang  together  in  masses  like  a 
membrane,  when  they  have  been  released  from  their  attachment  in  anatomical 
preparations.  On  reviewing  the  whole  arrangement,  there  can  be  no  doubt  that 
Corti's  oryau  is  an  apparatus  adapted  for  receiving  the  vibrations  of  the  membranu  II 


basilaris,  and  for  vibrating  of  itself,  but  our  present  knowledge  is  not  sufficient  toll 
determine  with  accuracy  the  manner  in  which  these  vibrations  take  place.  For 
this  purpose  we  require  to  estimate  the  stability  of  the  several  parts  and  the  degree 
of  tension  and  flexibility,  with  more  precision  than  can  be  deduced  from  such 
observations  as  have  hitherto  been  made  on  isolated  parts,  as  they  casually  group 
themselves  under  the  microscope. 

Now  Corti's  fibres  are  wound  round  and  covered  over  with  a  multitude  of  very 
delicate,  frail  formations,  fibres  and  cells  of  various  kinds,  partly  the  finest  ter- 
minational  runners  of  nerve  fibres  with  appended  nerve  cells,  partly  fibres  of  liga- 
mentous tissue,  which  appear  to  serve  as  a  support  for  fixing  and  suspending  the 
nerve  formations. 

The  connection  of  these  parts  is  best  shewn  in  fig.  49  opposite.  They  are 
grouped  like  a  pad  of  soft  cells  on  each  side  of  and  within  Corti's  arches.  The 
most  important  of  them  appear  to  be  the  cells  c  and  d,  which  arc  furnished  with 


142  DAMPING  OF  THE  VIBRATIONS  IN  THE  EAR.  part  i. 

hairs,  precisely  resembling  the  ciliated  colls  in  the  ampullae  and  utricidus.  They 
appear  to  be  directly  connected  by  fine  varicose  nerve  fibres,  and  constitute  the 
most  constant  part  of  the  cochlean  formations ;  for  with  birds  and  reptiles,  where 
the  structure  of  the  cochlea  is  much  simpler,  and  even  Corti's  arches  are  absent, 
these  little  ciliated  cells  are  always  to  be  found,  and  their  hairs  are  so  placed  as  to 
strike  against  Corti's  membrane  during  the  vibration  of  the  membrana  basilaris. 
The  cells  at  a  and  a',  fig.  49  (p.  140),  which  appear  in  an  enlarged  condition  at  b 
and  n  in  fig.  51  (p.  141),  seem  to  have  the  character  of  an  epithelium.  In  fig.  51 
there  will  also  be  observed  bundles  and  nets  of  fibres,  which  may  be  partly  merely 
supporting  fibres  of  a  ligamentous  nature,  and  may  partly,  to  judge  by  their  appear- 
ance as  strings  of  beads,  possess  the  character  of  bundles  of  the  finest  fibriles  of 
nerves.  But  these  parts  are  all  so  frail  and  delicate  that  there  is  still  much 
doubt  as  to  their  connection  and  office. 

^  The  essential  result  of  our  description  of  the  ear  may  consequently  be  said  to 
consist  in  having  found  the  terminations  of  the  auditory  nerves  everywhere  con- 
nected with  a  peculiar  auxiliary  apparatus,  partly  elastic,  partly  firm,  which  may  be 
put  in  sympathetic  vibration  under  the  influence  of  external  vibration,  and  will  then 
probably  agitate  and  excite  the  mass  of  nerves.  Now  it  was  shewn  in  Chap.  III., 
that  the  process  of  sympathetic  vibration  was  observed  to  difter  according  as  the 
bodies  put  into  sympathetic  vibration  were  such  as  when  once  put  in  motion  con- 
tinued to  sound  for  a  long  time,  or  soon  lost  their  motion,  p.  39c.  Bodies  which, 
like  tuning  forks  when  once  struck,  go  on  sounding  for  a  long  time,  are  susceptible 
of  sympathetic  vibration  in  a  high  degree  notwithstanding  the  difficulty  of  putting 
their  mass  in  motion,  because  they  admit  of  a  long  accumulation  of  impulses  in 
themselves  minute,  produced  in  them  by  each  separate  vibration  of  the  exciting 
tone.  But  precisely  for  this  reason  there  must  be  the  exactest  agreement  between 
the  pitches  of  the  proper  tone  of  the  fork  and  of  the  exciting  tone,  because  other- 

U  wise  subsequent  impulses  given  by  the  motion  of  the  air  could  not  constantly  recur 
in  the  same  phase  of  vibration,  and  thus  be  suitable  for  increasing  the  subsequent 
effect  of  the  preceding  impulses.  On  the  other  hand  if  we  take  bodies  for  which 
the  tone  rapidly  dies  away,  such  as  stretched  membranes  or  thin  light  strings,  we 
find  that  they  are  not  only  susceptible  of  sympathetic  vibration,  when  vibrating 
air  is  allowed  to  act  on  them,  but  that  this  sympathetic  vibration  is  not  so  limited 
to  a  particular  pitch,  as  in  the  other  case,  and  they  can  therefore  be  easily  set  in 
motion  by  tones  of  different  kinds.  For  if  an  elastic  body  on  being  once  struck 
and  allowed  to  sound  freely,  loses  nearly  the  whole  of  its  motion  after  ten  vibra- 
tions, it  will  not  be  of  much  importance  that  any  fresh  impulses  received  after  the 
expiration  of  this  time,  should  agree  exactly  with  the  former,  although  it  would  be 
of  great  importance  in  the  case  of  a  sonorous  body  for  which  the  motion  generated 
by  the  first  impulse  would  remain  nearly  unchanged  up  to  the  time  that  the  second 
impulse  was  applied.     In  the  latter  case  the  second  impulse  could  not  increase  the 

H  amount  of  motion,  unless  it  came  upon  a  phase  of  the  vibration  which  had 
precisely  the  same  direction  of  motion  as  itself. 

The  connection  between  these  two  relations  can  be  calculated  independently  of 
the  nature  of  the  body  put  into  sympathetic  vibration,*  and  as  the  results  are  iru- 
portant  to  enable  us  to  form  a  judgment  on  the  state  of  things  going  on  in  the  ear, 
a  short  table  is  annexed.  Suppose  that  a  body  which  vibrates  sympathetically  has 
been  set  into  its  state  of  maximum  vibration  by  means  of  an  exact  unison,  and 
that  the  exciting  tone  is  then  altered  till  the  sympathetic  vibration  is  reduced  to 
j^o  of  its  former  amount.  The  amount  of  the  required  difference  of  pitch  is  given  in 
the  first  column  in  terms  of  an  equally  tempered  Tone  [which  is  i  of  an  (Octave]. 
Now  let  the  same  sonorous  body  be  struck,  and  let  its  sound  be  allowed  to  die 
away  gradually.  The  number  of  vibrations  which  it  has  made  by  the  time  that  its 
intensity  is  reduced  to  ^\  of  its  original  amount  is  noted,  and  given  in  the  second 
column. 

*  The  mode  of  calculation  is  explained  in  Appendix  X. 


DAMPING  OF  THE  VIBRATIONS  IN  THE  EAR.  U3 


Difference  of  Pitch,  i:i  terms  of  an  equally  tempered  Tone,  iKHes- 

sary  to  reduce  the  intensity  of  sympathetic  vibration  to  ^'^  of  th  it 

produced  by  perfect  unisonance 

Number  of  vibrations  after  wliicli 
the  intensity  of  tone  in  a  sonorous 
body  whose  sound  is  allowed  to  die 
out,  reduces  to  ^'^   of  its  original 
amount 

1.  One  eighth  of  a  Tone 

2.  One  quarter  of  a  Tone 

3.  One  Semitone          ........ 

i.  Three  quarters  of  a  Tone 

5.  A  whole  Tone 

6.  A  Tone  and  a  quarter       ....... 

7.  A  tempered  minor  Third  or  a  Tone  and  a  half 

8.  A  Tone  and  three  quarters 

9.  A  tempered  major  Third  or  two  whole  Tones . 

38-00 
19-00 
9-50 
6-33 
4-75 
3-80 
3-17 
2-71 
2-37 

Now,  although  we  are  not  able  exactly  to  discover  how  long  the  ear  and  its 
individual  parts,  when  set  in  motion,  will  continue  to  sound,  yet  well-known  H 
•experiments  allow  us  to  form  some  sort  of  judgment  as  to  the  position  which 
the  parts  of  the  ear  must  occupy  in  the  scale  exhibited  in  this  table.  Thus,  there 
■cannot  possibly  be  any  parts  of  the  ear  which  continue  to  sound  so  long  as  a 
tuning-fork,  for  that  would  be  patent  to  the  commonest  observation.  But  even  if 
there  were  any  parts  in  the  ear  answering  to  the  first  degree  of  our  table,  that  is 
requiring  38  vibrations  to  be  reduced  to  ^\j  of  their  force, — we  should  recognise 
this  in  the  deeper  tones,  because  38  vibrations  last  i  of  a  second  for  A,  i  for  a, 
yW  for  a,  ifec,  and  such  a  long  endurance  of  sensible  sound  would  render  rapid 
musical  passages  impossible  in  the  unaccented  and  once-accented  Octaves.  Such 
41  state  of  things  would  disturb  musical  effect  as  much  as  the  sti'ong  resonance  of 
■a  vaulted  room,  or  as  raising  the  dampers  on  a  piano.  When  making  a  shake,  we 
■can  readily  strike  8  or  10  notes  in  a  second,  so  that  each  tone  separately  is  struck 
from  4  to  5  times.  If,  then,  the  sound  of  the  first  tone  had  not  died  off  in  our  ear 
before  the  end  of  the  second  sound,  at  least  to  such  an  extent  as  not  to  be  sensible  U 
when  the  latter  was  sounding,  the  tones  of  the  shake,  instead  of  being  individually 
distinct,  would  merge  into  a  continuous  mixture  of  both.  Now  shakes  of  this  kind, 
with  10  tones  to  a  second,  can  be  clearly  and  sharply  executed  throughout  almost 
the  whole  scale,  although  it  must  be  owned  that  from  A  downwards,  in  the  gi'eat 
and  contra  Octaves  they  sound  bad  and  rough,  and  their  tones  begin  to  mix.  Yet 
it  can  be  easily  shewn  that  this  is  not  due  to  the  mechanism  of  the  instrument. 
Thus  if  we  execute  a  shake  on  the  harmonium,  the  keys  of  the  lower  notes  are 
just  as  acc\u-ately  constructed  and  just  as  easy  to  move  as  those  of  the  higher 
ones.  Each  sepai-ate  tone  is  completely  cut  off  with  perfect  certainty  at  the 
iiioment  the  valve  falls  on  the  air  passage,  and  each  speaks  at  the  moment  the  valve 
is  raised,  because  during  so  brief  an  interruption  the  tongues  remain  in  a  state  of 
vibration.  Similarly  for  the  violoncello.  At  the  instant  when  the  finger  which 
makes  the  shake  falls  on  the  string,  the  latter  must  commence  a  vibration  oi  a 
different  periodic  time,  due  to  its  length ;  and  the  instant  that  the  finger  is  H 
removed,  the  vibration  belonging  to  the  deeper  tone  must  return.  And  yet  the 
skake  in  the  bass  is  as  imperfect  on  the  violoncello  as  on  any  other  instrument. 
Runs  and  shakes  can  be  relatively  best  executed  on  a  ])ianoforte  because,  at  the 
moment  of  striking,  the  new  tone  sounds  with  great  but  rapidly  decreasing  inten- 
sity. Hence,  in  addition  to  the  inharmonic  noise  produced  by  the  simultaneous 
continuance  of  the  two  tones,  we  also  hear  a  distinct  prominence  given  to  each 
separate  tone.  Now,  since  the  difficulty  of  shaking  in  the  bass  is  the  same  for  all 
instruments,  and  for  individual  instruments  is  demonstrably  independent  of  the 
manner  in  which  the  tones  are  produced,  we  are  forced  to  conclude  that  the 
difficulty  lies  in  the  ear  itself.  We  have,  then,  a  plain  indication  that  the  vibrating 
parts  of  the  ear  are  not  damped  with  sufficient  force  and  rapidity  to  allow  of 
vsuccessfully  effecting  such  a  rapid  alternation  of  tones. 

Nay  more,  this  fact  further  proves  that  there  must  he  (liferent  piu-ts  of  th'-  >'<ir 
which  are  set  in   vibration  by  tones  of  different  piti-lc  and  which  receive  the  sensatioa 


144  I)AMPIN(J  OF  THE  VIBHA.TIONS  IN  THE  EAR.  part  i. 

of  these  tones.  Thus,  it  might  be  supposed  that  as  the  vibratory  mass  of  the  whole 
ear,  the  drumskin,  auditory  ossicles,  and  fluid  in  the  labyrinth,  were  vibrating  at 
the  same  time,  the  inertia  of  this  mass  was  the  cause  why  the  sonorous  vibrations 
in  the  ear  were  not  immediately  extinguished.  But  this  hypothesis  would  not 
sufiice  to  explain  the  fact  observed.  For  an  elastic  body  set  into  sympathetic 
vibration  by  any  tone,  vibrates  sympathetically  in  the  pitch  number  of  the  exciting 
tone ;  but  as  soon  as  the  exciting  tone  ceases,  it  goes  on  sounding  in  the  pitch 
number  of  its  own  proper  tone.  This  fact,  which  is  derived  from  theory,  may  be 
perfectly  verified  on  tuning-forks  by  means  of  the  vibration  microscope. 

If,  then,  the  ear  vibrated  as  a  single  system,  and  were  capable  of  continuing 
its  vibration  for  a  sensible  time,  it  would  have  to  do  so  with  its  own  pitch  number, 
which  is  totally  independent  of  the  pitch  number  of  the  former  exciting  tone. 
The   consequence  is   that   shakes  would   be   eqiially  difficult   upon  both  high  and 

^  low  tones,  and  next  that  the  two  tones  of  the  shake  would  not  mix  with  each 
other,  but  that  each  Avould  mix  with  a  third  tone,  di;e  to  the  ear  itself.  We  became 
acquainted  with  such  a  tone  in  the  last  chapter,  the  high/""",  p.  116(r..  The  result, 
then,  under  these  circumstances  would  be  quite  different  from  what  is  observed. 

Now  if  a  shake  of  10  notes  in  a  second,  be  made  on  A,  of  which  the  vibra- 
tional number  is  110,  this  tone  would  be  struck  every  4  of  a  second.  We  may 
justly  assume  that  the  shake  would  not  be  clear,  if  the  intensity  of  the  expiring 
tone  were  not  reduced  to  y^  of  its  original  amount  in  this  i  of  a  second.  In  this 
case,  after  at  least  22  vibrations,  the  parts  of  the  ear  which  vibrate  sympathetically 
with  A  must  descend  to  at  least  ~  of  their  intensity  of  vibration  as  their  tone 
expires,  so  that  their  power  of  sympathetic  vibration  cannot  be  of  the  first  degree 
in  the  table  on  p.  143a,  but  may  belong  to  the  second,  third,  or  some  other  higher 
degree.  That  the  degree  cannot  be  any  much  higher  one,  is  shewn  in  the  first 
place  by  the  fact  that  shakes  and  runs  begin  to  be  difficult  even  on  tones  which  do 

^  not  lie  much  lower.  This  we  shall  see  by  observations  on  beats  subsequently  de- 
tailed. We  may  on  the  whole  assume  that  the  parts  of  the  ear  which  vibrate 
sympathetically  have  an  amount  of  damping  power  corresponding  to  the  third 
degree  of  our  table,  where  the  intensity  of  sympathetic  vibration  with  a  Semitone 
difference  of  pitch  is  only  ^^  of  what  it  is  for  a  complete  unison.  Of  course  there 
can  be  no  question  of  exact  determinations,  but  it  is  important  for  us  to  be  able 
to  form  at  least  an  approximate  conception  of  the  influence  of  damping  on  the 
sympathetic  vibration  of  the  ear,  as  it  has  great  significance  in  the  relations 
of  consonance.     Hence  when  we   hereafter    speak  of    individual  parts  of  the   ear 

!  vibrating  sympathetically  with  a  determinate  tone,  we  mean  that  they  are  set  into 
strongest  motion  by  that  tone,  but  are  also  set  into  vibration  less  strongly  by  tones 
of  nearly  the  same  pitch,  and  that  this  sympathetic  vibration  is  still  sensible  for 
the  interval  of  a  Semitone.  Fig.  52  may  serve 
to  give  a  general  conception  of  the  law  by  which 

^the  intensity  of  the  sympathetic  vibration  de- 
creases, as  the  difference  of  pitch  increases.  The 
horizontal  line  a  b  c  represents  a  portion  of  the 
musical  scale,  each  of  the  lengths  a  b  and  b  c 
standing  for  a  whole  (equally  tempered)  Tone. 
Suppose  that  the  body  which  vibrates  sympa- 
thetically has  been  tuned  to  the  tone  b  and  that 
the    vertical   line   b  d  represents  the    maxinuun 

of  intensity  of  tone  which  it  can  attain  when  excited  by  a  tone  in  perfect  unison 
with  it.  On  the  base  line,  intervals  of  y\j  of  a  whole  Tone  are  set  off",  and  the  ver- 
tical lines  drawn  through  them  shew  the  corresponding  intensity  of  the  tone  in  the 
body  which  vibrates  sympathetically,  when  the  exciting  tone  differs  from  a  unison 
by  the  corresponding  interval.  The  following  are  the  numbers  from  which  fig.  52 
was  constructed  : — 


CHAP.  VI.         THEORY  OF  THE  FUNCTION  OF  THE  COCHLEA. 


145 


Difference  of  Pitch 

Intensity 
of  .Sympathetic  \'ibrati<)n 

DiffeieiKi'  (if  Pitch 

Intensity 
i.f  Synipatlietic  Vibration 

00 

100 

0-(J 

7-2 

0-1 

74 

0-7 

5-4 

0-2 

41 

0-8 

4-2 

0-3 

24 

0-9 

3-3 

0-4 

15 

Whole  Tone 

2-7 

Semitone 

10 

Now  we  cannot  precisely  ascertain  what  parts  of  the  ear  actually  vibrate  sym- 
pathetically with  individual  tones.*  We  can  only  conjecture  what  they  are  at 
present  in  the  case  of  human  beings  and  mammals.  The  whole  construction  of 
the  partition  of  the  cochlea,  and  of  Corti's  arches  which  rest  upon  it,  appears  most 
suited  for  executing  independent  vibrations.  We  do  not  need  to  require  of  them 
tlie  power  of  continuing  their  vibrations  for  a  long  time  without  assistance.  ^ 

But  if  these  formations  are  to  serve  for  distinguishing  tones  of  different  pitch, 
and  if  tones  of  difterent  pitch  are  to  be  equally  well  perceived  in  all  parts  of  the 
scale,  the  elastic  formations  in  the  cochlea,  which  are  connected  with  different 
nerve  fibres,  miist  be  differently  tuned,  and  their  proper  tones  must  form  a  regu- 
larly progressive  series  of  degrees  through  the  whole  extent  of  the  musical  scale. 

According  to  the  recent  anatomical  researches  of  V.  Hensen  and  C.  Hasse,  it 
is  probably  the  breadth  of  the  membrana  basilaris  in  the  cochlea,  which  deter- 
mines the  tuning. t  At  its  commencement  opposite  the  oval  window,  it  is 
comparatively  narrow,  and  it  continually  increases  in  width  as  it  approaches  the 
a})ex  of  the  cochlea.  The  following  measurements  of  the  membrane  in  a  newly 
born  child,  from  the  line  where  the  nerves  pass  through  on  the  inner  edge,  to  the 
attachment  to  the  ligamentum  spirale  on  the  outer  edge,  are  given  by  V.  Hensen : — 


Place  of  Section 

Breadth  of  Membrane  or  Length  of  Trans- 
verse Fibres. 

Millimetres 

Inches 

0-2625  mm.  [  =  0-010335  in.]  from  root     . 
0-8626  mm.  [=0-033961  in.]  from  root     . 

Middle  of  the  first  spire 

End  of  first  spire 

Middle  of  second  spire 

End  of  second  spire           ...... 

At  the  hamulus          ....... 

0-04125 

0-0825 

0-169 

0-3 

0-4125 

0-45 

0-495 

•00162 
-00325 
•00665 
•01181 
•01624 
•01772 
•01949 

The  breadth  therefore  increases  more  than  twelvefold  from  the  beginning  to 
the  end. 

Corti's  rods  also  exhibit  an  increase  of  size  as  they  approach  the  vertex  of  the 
cochlea,  but  in  a  much  less  degree  than  the  membrana  basilaris.  The  following 
are  Hensen's  measurements  . — 


at  the  round  window 

at  the  hamulus 

mm. 

inch 

mm. 

inch 

Length  of  inner  rod  .... 
Length  of  outer  rod  .... 
Span  of  the  arch         .... 

0-048 
0-048 
0-019 

0-00189 
0-00189 
0-00075 

0-0855 

0-098 

0-085 

0-00337 
0-0038G 
0-00335 

*  [Here  the  passage,  '  The  particles  of 
auditory  sand,'  to  'used  for  musical  tones,' 
on  pp.  217-18  of  the  1st  English  edition  has 
been  cancelled,  and  the  passage  '  We  can  only 
-conjecture,'  to  '  without  assistance,'  on  p.  145a 
added  in  its  place  from  the  4th  German  edition. 
—  Translator.^ 

tin  the  1st  [German]  edition  of  this  book 
(1863),  which  was  written  at  a  time  when  the 
more  delicate  anatomy  of  the  cochlea  was  just 


beginning  to  be  developed,  I  supposed  that  the 
different  degrees  of  stiffness  and  tension  in 
Corti's  rods  themselves  might  furnish  the 
reason  of  their  different  tuning.  By  Hensen's 
measures  of  the  breadth  of  the  membrana 
basilaris  {Zeitschrift  fiir  wissensch.  Zoologie, 
vol.  xiii.  p.  492)  and  Hasse's  proof  that  -Corti's 
rods  are  absent  in  birds  and  amphibia,  far  more 
definite  foundations  for  forming  a  judgment 
have  been  furnished,  than  I  then  possessed. 

L 


146  THEORY  UF  THE  FUNCTION  OF  THE  COCHLEA.  part  i. 

Hence  it  follows,  as  Henle  has  also  proved,  that  the  greatest  increase  of  breadth 
falls  on  the  outer  zone  of  the  basilar  membrane,  beyond  the  line  of  the  attach- 
ment of  the  outer  rods.  This  increases  from  0-023  mm.  [=  -000905  in.]  to  0-41 
mm.  [=  -016142  inch]  or  neai'ly  twentyfold. 

In  accordance  with  these  measures,  the  two  rows  of  Corti's  rods  are  almost 
parallel  and  upright  near  to  the  round  window,  but  they  are  bent  much  more 
strongly  towards  one  another  near  the  vertex  of  the  cochlea. 

It  has  been  already  mentioned  that  the  membrana  basilaris  of  the  cochlea 
breaks  easily  in  the  radial  direction,  but  that  its  radial  fibres  have  considerable 
tenacity.  This  seems  to  me  to  furnish  a  very  important  mechanical  relation, 
namely,  that  this  membrane  in  its  natural  connection  admits  of  being  tightly 
stretched  in  the  transverse  direction  from  the  modiolus  to  the  outer  wall  of  the 
cochlea,  but  can  have  only  little  tension  in  the  direction  of  its  length,  because  it 
^  could  not  resist  a  strong  pull  in  this  direction. 

Now  the  mathematical  theory  of  the  vibration  of  a  membrane  with  different  ten- 
sions in  different  directions  shews  that  it  behaves  very  differently  from  a  membrane 
which  has  the  same  tension  in  all  directions.*  On  the  latter,  vibrations  produced 
in  one  part,  spread  uniformly  in  all  directions,  and  hence  if  the  tension  were  uniform 
it  would  be  impossible  to  set  one  part  of  the  basilar  membrane  in  vibration,  without 
producing  nearly  as  strong  vibrations  (disregarding  individual  nodal  lines)  in  all  other 
parts  of  the  membrane. 

But  if  the  tension  in  direction  of  its  length  is  infinitesimally  small  in  com- 
parison with  the  tension  in  direction  of  the  breadth,  then  the  radial  fibres  of 
the  basilar  membrane  may  be  approximatively  regarded  as  forming  a  system  of 
stretched  strings,  and  the  membranous  connection  as  only  serving  to  give  a  ful- 
crum to  the  pressure  of  the  fluid  against  these  strings.  In  that  case  the  laws  of 
their  motion  would  be  the  same  as  if  every  individual  string  moved  independently 
H  of  all  the  others,  and  obeyed,  by  itself,  the  influence  of  the  periodically  alternating 
pressure  of  the  fluid  of  the  labyrinth  contained  in  the  vestibule  gallery.  Conse- 
quently any  exciting  tone  would  set  that  part  of  the  membi-ane  into  sympathetic 
vibration,  for  which  the  proper  tone  of  one  of  its  radial  fibres  that  are  stretched 
and  loaded  with  the  various  appendages  already  described,  corresponds  most  nearly 
with  the  exciting  tone ;  and  thence  the  vibrations  will  extend  with  rapidly  dimi- 
nishing strength  on  to  the  adjacent  parts  of  the  membrane.  Fig.  52,  on  p.  144(/, 
might  be  taken  to  represent,  on  an  exaggerated  scale  of  height,  a  longitudinal  sec- 
tion of  that  part  of  the  basilar  membrane  in  which  the  proper  tone  of  the  radial 
fibres  of  the  membrane  are  nearest  to  the  exciting  tone. 

The  strongly  vibrating  parts  of  the  membrane  would,  as  has  been  explained  in 
respect  to  all  bodies  which  vibrate  sympathetically,  be  more  or  less  limited,  accord- 
ing to  the  degree  of  damping  power  in  the  adjacent  parts,  by  friction  against  the 
fluid  in  the  labyrinth  and  in  the  soft  gelatinous  parts  of  the  nerve  fillet. 
51  Under  these  circumstances  the  parts  of  the  membrane  in  unison  with  higher 
tones  must  be  looked  for  near  the  round  window,  and  those  with  the  deeper,  near 
the  vertex  of  the  cochlea,  as  Hensen  also  concluded  from  his  measurements.  That 
such  short  strings  should  be  capable  of  corresponding  with  such  deep  tones,  must 
be  explained  by  their  being  loaded  in  the  basilar  membrane  with  all  kinds  of  solid 
formations ;  the  fluid  of  both  galleries  in  the  cochlea  must  also  be  considered  as 
weighting  the  membrane,  because  it  cannot  move  without  a  kind  of  wave  motion 
in  that  fluid. 

The  observations  of  Hasse  shew  that  Corti's  arches  do  not  exist  in  the  cochlea 
of  birds  and  amphibia,  although  the  other  essential  parts  of  the  cochlea,  as  the 
basilar  membrane,  the  ciliated  cells  in  connection  with  the  terminations  of  the 
nerves,  and  Corti's  membrane,  which  stands  opposite  the  ends  of  these  ciliae,  are 
all  present.  Hence  it  becomes  very  probable  that  Corti's  arches  play  only  a 
secondary  part  in  the  function  of  the  cochlea.  Perhaps  we  might  look  for  the  effect 
*  See  Appendix  XI. 


CHAP.  VI.  THEORY  OF  THE  FUNCTION  OF  THE  COCHLE.V.  117 

of  Corti's  arches  in  their  power,  as  relatively  firm  objects,  of  trausuiittinu'  the 
vibrations  of  the  basilar  membrane  to  small  limited  regions  of  the  iii)per  part  of 
the  relatively  thick  nervons  fillet,  better  than  it  conld  be  done  by  the  immediate 
communication  of  the  vibrations  of  the  basilar  membrane  tlu'ough  the  soft  mass 
of  this  fillet.  Close  to  the  outside  of  the  upper  end  of  the  ai'ch,  connected  with 
it  by  the  stiflFer  fibriles  of  the  membrana  reticularis,  are  the  ciliated  cells  of  tlie 
nervous  fillet  (see  c  in  fig.  49,  p.  140).  In  birds,  on  the  other  hand,  the  ciliated  cells 
form  a  thin  stratum  upon  the  basilar  naembrane,  and  this  stratum  can  readily 
receive  limited  vibrations  from  the  membrane,  without  communicating  them  too 
far  sideways. 

According  to  this  view  Corti's  arches,  in  the  last  resort,  will  be  the  means  of 
transmitting  the  vibrations  received  from  the  basilar  membrane  to  the  terminal 
appenaages  of  the  conducting  nerve.  In  this  sense  the  reader  is  requested  here- 
after to  understand  references  to  the  vibrations,  proper  tone,  and  intonation  ofH 
Corti's  arches  ;  the  intonation  meant  is  that  which  they  receive  through  their 
connection  with  the  corresponding  part  of  the  basilar  membrane. 

According  to  Waldeyer  there  are  about  4500  outer  arch  fibres  in  the  human 
cochlea.  If  we  deduct  300  for  the  simple  tones  which  lie  beyond  musical  limits, 
and  cannot  have  their  pitch  perfectly  apprehended,  there  remain  4200  for  the 
seven  octaves  of  musical  instiiiments,  that  is,  600  for  every  Octave,  50  for  every 
Semitone  (that  is,  1  for  every  2  cents) ;  certainly  quite  enough  to  explain  the 
power  of  distinguishing  small  pai-ts  of  a  Semitone.*  According  to  Prof.  W. 
Preyer's  investigations,+  practised  musicians  can  distinguish  with  certainty  a 
diflference  of  pitch  arising  from  half  a  vibration  in  a  second,  in  the  doubly 
accented  Octave.  This  would  give  1000  distinguishable  degrees  of  pitch  in  the 
Octave,  from  500  to  1000  vibrations  in  the  second.  Towards  the  limits  of  the 
scale  the  power  to  distinguish  differences  diminishes.  The  4200  Corti's  arches 
appear  then,  in  this  respect,  to  be  enough  to  apprehend  distinctions  of  thisH 
amount  of  delicacy.  But  even  if  it  should  be  found  that  many  more  than 
4200  degrees  of  pitch  could  be  distinguished  in  the  Octave,  it  would  not  prejudice 
our  assumption.  For  if  a  simple  tone  is  struck  having  a  pitch  between  those  of 
two  adjacent  Corti's  arches,  it  would  set  them  both  in  sympathetic  vibration,  and 
that  arch  would  vibrate  the  more  strongly  which  was  nearest  in  pitch  to  the 
proper  tone.  The  smallness  of  the  interval  between  the  pitches  of  two  fibres  still 
distinguishable,  will  therefore  finally  depend  upon  the  delicacy  with  Avhich  the 
different  forces  of  the  vibrations  excited  can  be  compared.  And  we  have  thus 
also  an  explanation  of  the  fact  that  as  the  pitch  of  an  external  tone  rises  con- 
tinuously, our  sensations  also  alter  continuonsly  and  not  by  jumps,  as  must  be  the 
case  if  only  one  of  Corti's  arches  were  set  in  sympathetic  motion  at  once. 

To  draw  further  conclusions  from  our  hypothesis,   when  a  simple  tone  is  pre- 
sented to  the  ear,  those  Corti's  arches   which  are  nearly  or  exactly  in  unison  with 
it  will  be  strongly  excited,  and  the  rest   only  slightly  or  not  at  all.     Hence  every  H 
simple    tone   of  determinate  pitch  will   be  felt  only  by  certain  nerve  fibres,   and 

*  [A  few  lines  of  the  1st  English  edition  at  vib.       a  difference  of    or  interval  of 

have  here   been   cancelled,    and  replaced   by  500              -300  vib.            1-0  cents )  ?/'as  per- 

others  from  the  4th  German  edition,— rrrms-  1000              -500    „                -9      „      jceived. 

^f'tor.]  but  on  the  other  hand 

\  \_Ueher     die     Grenzen     der     Tomvahrneh-  at  vib.       a  difference  of    or  interval  of 

mimg  (On  the   limits   of   the    perception    of  60              -200  vib.                 6  cents \   was 

tone),  June  1876.     Eearranged  in  English  by  HO             -091    ,,                1-4      „      |   not 

the    Translator    in    the     Proceedings    of    the  250             -150    ,,                1-0      „      ["  per- 

Miisical    Association     for    1876-7,    pp.     1-32,  400              -200    ,,                  -9      „     )   ceived, 

under  the  title  of  '  On  the  Sensitiveness  of  the  the  intervals  perceived,  or  not  perceived,  being 

Ear  to  Pitch  and  Change  of  Pitch  in  Music'  the  same,  but  the  pitches  different.    And  gene- 

On  p.  11  of  this  arrangement  it  is  stated  that,  rally  throughout  the  scale  a  difference  of   i 

including  Delezenne's  results,  vib.  is  not  heard,  but 

at  vib.        a  difference  of    or  interval  of  from  Gj^  to  j/j^  f  vib.^ 

120             -418  vib.               6  cents  )  i««s  per-  and  from      «'  to    c"  I    „    Ws  heard. 

440             -364    „             1-4     ,,     jceived.  andfrom       c  to    c"  ^    „  J 

—  Traiislator.] 
L  2 


148  THEORY  OF  THE  FUNCTION  OF  THE  COCHLEA.  part  i. 

simple  tones  of  difterent  pitch  will  excite  different  fibres.  When  a  compound 
musical  tone  or  chord  is  presented  to  the  ear,  all  those  elastic  bodies  will  be 
excited,  which  have  a  proper  pitch  corresponding  to  the  various  individual  simple 
tones  contained  in  the  whole  mass  of  tones,  and  hence  by  properly  directing 
attention,  all  the  individual  sensations  of  the  individual  simple  tones  can  be 
perceived.  The  chord  must  be  resolved  into  its  individual  compound  tones,  and 
the  compound  tone  into  its  individual  harmonic  partial  tones. 

This  also  explains  how  it  is  that  the  ear  resolves  a  motion  of  the  air  into 
pendular  vibrations  and  no  other.  Any  particle  of  air  can  of  course  execute  only 
one  motion  at  one  time.  That  we  considered  such  a  motion  mathematically  as  a 
sum  of  pendular  vibrations,  w^as  in  the  first  instance  merely  an  arbitrary  assump- 
tion to  facilitate  theory,  and  had  no  meaning  in  nature.  The  first  meaning  in 
nature  that  we  found  for  this  resolution  came  from  considering  sympathetic 
51  vibration,  when  we  discovered  that  a  motion  which  was  not  pendular,  could 
produce  sympathetic  vibrations  in  bodies  of  those  difterent  pitches,  which  cor- 
responded to  the  harn)onic  upper  partial  tones.  And  now^  our  hypothesis  has  also 
reduced  the  phenomenon  of  hearing  to  that  of  sympathetic  vibration,  and  thus 
furnished  a  reason  why  an  originally  simple  periodic  vibration  of  the  air  pro- 
duces a  sum  of  different  sensations,  and  hence  also  appears  as  compound  to  our 
perceptions. 

Tlie  sensation  of  difterent  pitch  would  consequently  be  a  sensation  in  different 
nerve  fibres.  The  sensation  of  a  quality  of  tone  would  depend  upon  the  power  of 
a  given  compound  tone  to  set  in  vibration  not  only  those  of  Corti's  arches  which 
correspond  to  its  prime  tone,  but  also  a  series  of  other  arches,  and  hence  to  excite 
sensation  in  several  different  groups  of  nerve  fibres. 

Physiologically  it  should  be  observed  that  the  present  assumption  reduces 
sensations  which  differ  qualitatively  according  to  pitch  and  quality  of  tone,  to  a 
5j  difference  in  the  nerve  fibres  which  are  excited.  This  is  a  ste^J  similar  to  that 
taken  in  a  wider  field  by  Johannes  Miiller  in  his  theory  of  the  specific  energies  of 
sense.  He  has  shewn  that  the  difference  in  the  sensations  due  to  various  senses, 
does  not  depend  upon  the  actions  which  excite  them,  but  upon  the  various  nervous 
arrangements  which  receive  them.  We  can  convince  ourselves  experimentally 
that  in  whatever  manner  the  optic  nerve  and  its  expansion,  the  retina  of  the  eye, 
may  be  excited,  b}'  light,  by  twitching,  by  pressm-e,  or  by  electricit}',  the  result  is 
never  anything  but  a  sensation  of  light,  and  that  the  tactual  nerves,  on  the  contrary, 
never  give  us  sensations  of  light  or  of  hearing  or  of  taste.  The  same  solar  rays 
which  are  felt  as  light  by  the  eye,  are  felt  by  the  nerves  of  the  hand  as  heat ;  the 
same  agitations  which  are  felt  by  the  hand  as  twitterii:igs,  are  tone  to  the  ear. 

Just  as  the  ear  apprehends  vibrations  of  different  periodic  time  as  tones  of 
different  pitch,  so  does  the  eye  perceive  luminiferous  vibrations  of  different  periodic 
time  as  difterent  coloiu's,  the  quickest  giving  violet  and  blue,  the  mean  green  and 
II  yellow,  the  slowest  red.  The  laws  of  the  mixture  of  colours  led  Thomas  Young 
to  the  hypothesis  that  there  were  three  kinds  of  nerve  fibres  in  the  eye,  with 
different  powers  of  sensation,  for  feeling  red,  for  feeling  green,  and  for  feeling 
violet.  In  reality  this  assumption  gives  a  very  simple  and  perfectl}'  consistent 
explanation  of  all  the  optical  phenomena  depending  on  colour.  And  by  this  means 
the  qiialitative  differences  of  the  sensations  of  sight  are  reduced  to  differences  in 
the  nerves  which  receive  the  sensations.  For  the  sensations  of  each  individual 
fibre  of  the  optic  nerve  there  remains  only  the  quantitative  differences  of  greater  or 
less  irritation. 

The  same  result  is  obtained  for  hearing  by  the  hypothesis  to  wdiich  our 
investigation  of  quality  of  tone  has  led  us.  The  qualitative  difference  of  pitch 
and  quality  of  tone  is  reduced  to  a  difference  in  the  fibres  of  the  nerves  receiving 
the  sensation,  and  for  each  individual  fibre  of  the  nerve  there  remains  only  the 
quantitative  differences  in  the  amount  of  excitement. 

The  processes  of  irritation  within  the  nerves  of  the  muscles,  by  which  their 
contraction   is    determined,   have    hitherto    been    more    accessible    to    physiological 


CHAP.  VI.         THEORY  OF  THE  FUNCTION  OF  THE  COCHLEA.  140 

investigation  than  those  which  take  place  in  the  nerves  of  sense.  In  tliose  oi  the 
muscle,  indeed,  we  find  only  (inantitative  difterences  of  more  or  less  excitement, 
and  no  qualitative  differences  at  all.  In  them  we  are  able  to  establish,  that  during 
excitement  the  electrically  active  particles  of  the  nerves  undergo  determinate 
changes,  and  that  these  changes  ensue  in  exactly  the  same  way  whatever  be  the 
excitement  which  causes  them.  But  precisely  the  same  changes  also  take  place  in 
an  excited  nerve  of  sense,  although  their  consequence  in  this  case  is  a  sensation, 
while  in  the  other  it  was  a  motion  ;  and  hence  we  see  that  the  mechanism  of  the 
process  of  irritation  in  the  nerves  of  sense  must  be  in  every  respect  similar  to  that 
in  the  nerves  of  motion.  The  two  hypotheses  just  explained  really  red\ice  the 
processes  in  the  nerves  of  man's  two  principal  senses,  notwithstanding  their 
apparently  involved  qualitative  differences  of  sensations,  to  the  same  simple 
scheme  with  which  we  are  familiar  in  the  nerves  of  motion.  Nerves  have  been 
often  and  not  unsuitably  compared  to  telegraph  wires.  Such  a  wire  conducts  one  H 
kind  of  electric  current  and  no  other ;  it  may  be  stronger,  it  may  be  weaker,  it  may 
move  in  either  direction  ;  it  has  no  other  qualitative  differences.  Nevertheless, 
according  to  the  different  kinds  of  apparatus  with  wliich  we  provide  its  termina- 
tions, we  can  send  telegraphic  despatches,  ring  bells,  explode  mines,  decompose 
water,  move  magnets,  magnetise  iron,  develop  light,  and  so  on.  So  with  the 
nerves.  The  condition  of  excitement  which  can  be  produced  in  them,  and  is  con- 
ducted by  them,  is,  so  far  as  it  can  be  recognised  in  isolated  fibres  of  a  nerve, 
everywhere  the  same,  but  when  it  is  brought  to  various  parts  of  the  brain,  or 
the  body,  it  produces  motion,  secretions  of  glands,  increase  and  decrease  of  the 
quantity  of  blood,  of  redness  and  of  warmth  of  individual  organs,  and  also  sensa- 
tions of  light,  of  hearing,  and  so  forth.  Supposing  that  every  qualitatively 
different  action  is  produced  in  an  organ  of  a  different  kind,  to  which  also  separate 
fibres  of  nerve  must  proceed,  then  the  actual  process  of  irritation  in  individual 
nerves  may  always  be  precisely  the  same,  just  as  the  electrical  current  in  the  tele-  ^ 
graph  wires  remains  one  and  the  same  notwithstanding  the  various  kinds  of 
effects  which  it  produces  at  its  extremities.  On  the  other  hand,  if  we  assume  that 
the  same  fibre  of  a  nerve  is  capable  of  conducting  difterent  kinds  of  sensation,  we 
should  have  to  assume  that  it  admits  of  various  kinds  of  processes  of  irritation, 
and  this  we  have  been  hitherto  unable  to  establish. 

In  this  respect  then  the  view  here  proposed,  like  Young's  hypothesis  for  the 
difference  of  colours,  has  a  still  wider  signification  for  the  physiology"  of  the 
nerves  in  general. 

Since  the  first  publication  of  this  book,  the  theory  of  auditory  sensation  here 
explained,  has  received  an  interesting  confirmation  from  the  observations  and 
experiments  made  by  V.  Hensen*  on  the  auditory  apparatus  of  the  Crustaceae. 
These  animals  have  liags  of  auditory  stones  (otoliths),  partly  closed,  partly 
opening  outwards,  in  which  these  stones  float  freely  in  a  watery  fluid  and  are 
supported  by  hairs  of  a  peculiar  formation,  attached  to  the  stones  at  one  end,  and,  H 
partly,  arranged  in  a  series  proceeding  in  order  of  magnitude,  from  larger  and 
thicker  to  shorter  and  thinner.  In  many  crustaceans  also  we  find  precisely 
similar  hairs  on  the  open  surface  of  the  body,  and  these  must  be  considered  as 
auditory  hairs.  The  proof  that  these  external  hairs  are  also  intended  for  hearing, 
depends  first  on  the  similarity  of  their  construction  with  that  of  the  hairs  in  the 
bags  of  otoliths ;  and  secondly  on  Hensen's  discovery  that  the  sensation  of 
hearing  remained  in  the  Mysis  (opossiim  shrimp)  when  the  bags  of  otoliths  had 
been  extirpated,  and  the  external  auditory  hairs  of  the  antennae  were  left. 

Hensen  conducted  the  sound  of  a  keyed  Inigle  through  an  apparatus  formed  on 
the  model  of  the  drumskin  and  auditory  ossicles  of  the  ear  into  the  water  of  a 
little  box  in  which  a  specimen  of  Mysis  was  fastened  in  such  a  way  as  to  allow 
the  external  auditory  hairs  of  the  tail  to  be  observed.  It  was  then  seen  that 
certain  tones  of  the  horn  set  certain  hairs  into  strong  vibration,  and  other  tones 

*  Stiulicn  i'lhcr  das  Gehoron/aii  da  Decco-  and  Kolliker's  Zcitschrift  fiir  irlssensch/'/tlichc 
poden,  Leipzig,  1863.     Reprinted  from  Siebold       Zoolotjic,  vol.  xiii. 


150  THEORY  OF  THE  FUNCTION  OF  THE  COCHLEA.  part  i. 

other  hairs.  Each  hair  answered  to  several  notes  of  tlie  horn,  and  from  the 
notes  mentioned  we  can  approximatively  recognise  the  series  of  under  tones  of  one 
and  the  same  simple  tone.  The  results  could  not  be  ([uite  exact,  because  the 
resonance  of  the  conducting  apparatus  must  have  had  some  influence. 

Thus  one  of  these  hairs  answered  strongly  to  (/jf  and  d'^  more  weakly  to  g, 
and  very  weakly  to  G.  Tliis  leads  us  to  suppose  that  it  was  tuned  to  some  pitch 
between  d"  and  d"^.  In  that  case  it  answered  to  the  second  partial  of  d'  to  d'^ 
the  third  of  r/  to  gji,  the  fourth  of  d  to  cZ|,  and  the  sixth  of  G  to  G^.  A  second 
hair  answered  strongly  to  ajj;  and  the  adjacent  tones,  more  weakly  to  d^  and  A^. 
Its  proper  tone  therefore  seems  to  have  been  aj^. 

By  these  observations  (which  through  the  kindness  of  Herr  Hensen  I  have 
myself  had  the  opportunity  of  verifying)  the  existence  of  such  relations  as  we  have 
supposed  in  the  case  of  the  human  cochlea,  have  been  directly  proved  for  these 

^Crustaceans,  and  this  is  the  more  valuable,  because  the  concealed  position  and 
ready  destructibility  of  the  corresponding  organs  of  the  human  ear  give  little  hope 
of  our  ever  being  able  to  make  such  a  direct  experiment  on  the  intonation  of  its 
individual  parts.* 

So  far  the  theory  which  has  been  advanced  refers  in  the  first  place  only  to 
the  lasting  sensation  produced  by  regular  and  continued  periodical  oscillations. 
But  as  regards  the  2)erception  of  irregtdar  motions  of  the  air,  that  is,  of  noises,  it 
is  clear  that  an  elastic  apparatus  for  executing  vibrations  could  not  remain  at 
absolute  rest  in  the  presence  of  any  force  acting  upon  it  for  a  time,  and  even  a 
momentary  motion  or  one  recurring  at  irregular  intervals  would  suffice,  if  only 
powerful  enough,  to  set  it  in  motion.  The  peculiar  advantage  of  resonance  over 
proper  tone  depends  precisely  on  the  fact  that  disproportionately  weak  individual 
impulses,  provided  that  they  succeed  each  other  in  correct  rhythm,  are  capable  of 
producing   comparatively  considerable   motions.      On   the  other  hand,  momentary 

^  but  strong  impulses,  as  for  example  those  which  result  from  an  electric  spark,  will 
set  every  part  of  the  basilar  membrane  into  an  almost  equally  powerful  initial 
motion,  after  which  each  part  would  die  off  in  its  own  proper  vibrational  period. 
By  that  means  there  might  arise  a  simultaneous  excitement  of  the  whole  of  the 
nerves  in  the  cochlea,  which  although  not  equally  powerful  would  yet  be  propor- 
tionately gradated,  and  hence  could  not  have  the  character  of  a  determinate  pitch. 
Even  a  weak  impression  on  so  many  nerve  fibres  will  produce  a  clearer  impression 
than  any  single  impression  in  itself.  We  know  at  least  that  small  difi'erences  of 
brightness  are  more  readily  perceived  on  large  than  on  small  parts  of  the  circle  of 
vision,  and  little  differences  of  temperature  can  be  better  perceived  by  plunging 
tlie  whole  arm,  than  by  merely  dipping  a  finger,  into  the  warm  water. 

Hence  a  perception  of  momentary  impulses  by  the  cochlear  nerves  is  quite 
possible,  just  as  noises  are  perceived,  without  giving  an  especially  sensible  pro- 
minence to  any  determinate  pitch. 

^  If  the  pressure  of  the  air  which  bears  on  the  drumskin  lasts  a  little  longer,  it 
will  favour  the  motion  in  some  regions  of  the  basilar  membrane  in  preference  to 
other  parts  of  the  scale.  Certain  pitches  will  therefore  be  especially  prominent. 
This  we  may  conceive  thus  :  every  instant  of  pressure  is  considered  as  a  pressure 
that  wull  excite  in  every  fibre  of  the  basilar  membrane  a  motion  corresponding 
to  itself  in  direction  and  strength  and  then  die  off;  and  all  motions  in  each 
fibre  which  are  thus  excited  are  added  algebraically,  whence,  according  to  cir- 
cumstances, they  reinforce  or  enfeeble  each  other.t  Thus  a  uniform  jjressure 
which  lasts  during  the  first  half  vibration,  that  is,  as  long  as  the  first  positive 
cxcTU-sion,  increases  the  excursion  of  the  vibrating  body.  But  if  it  lasts  longer 
it  weakens  the  effect  first  produced.  Hence  rapidly  vibrating  bodies  would  be 
proportionably  less  excited  by  such  a  pressure,  than  those  for  which  half  a  vibra- 
tion lasts  as  long  as,  or  longer  than,  the  pressure  itself.     By  this  means  such  an 

*  [From  here  to  the  end  of  this  chapter  is  f  See  the  mathematical  expression  for  this 

au  addition  from  the  4th  German  edition.—       conception  at  the  end  of  Appendix  XI. 

'J'ranshttor.] 


CHAP.  VI.  THEORY  OF  THE   FUNCTION   OF  THE  (MK'HLEA.  If)! 

impression  would  acquire  a  certain,  though  an  ill-defined,  pitch,  in  general  the 
intensity  of  the  sensation  seems,  for  an  equal  amount  of  vis  viva  in  the  motion,  to 
increase  as  the  pitch  ascends.  So  that  the  impression  of  the  highest  strongly 
excited  fibre  preponderates. 

A  determinate  pitch,  to  a  more  remarkable  extent,  may  also  naturally  result,  if 
the  pressure  itself  which  acts  on  the  stirruj)  of  the  drum  alternates  several  times 
between  positive  and  negative.  And  thus  all  transitional  degrees  between  noises 
without  any  determinate  pitch,  and  compound  tones  with  a  determinate  pitch  may 
be  produced.  This  actually  takes  place,  and  herein  lies  the  proof,  on  which  Hen- 
S.  Exner  *  has  properly  laid  weight,  that  such  noises  must  be  perceived  by  those 
})arts  of  the  ear  which  act  in  distinguishing  pitch. 

In  former  editions  of  this  work  I  had  expressed  a  conjecture  that  the  auditory 
ciliae  of  the  ampullae,   which  seemed  to  be  but  little  adapted  for  resonance,  and 
those  of   the    little  bags  opposite  the    otoliths,  might  be  especially  active  in  the  11 
perception  of  noises. 

As  regards  the  ciliae  in  the  ampullae,  the  investigations  of  Goltz  have  made  it 
extremely  probable  that  they,  as  well  as  the  semicircular  canals,  serve  for  a  totally 
difterent  kind  of  sensation,  namely  for  the  perception  of  the  turning  of  the  head. 
Revolution  about  an  axis  perpendicular  to  the  plane  of  one  of  the  semicircular 
canals  cannot  be  immediately  transferred  to  the  ring  of  water  which  lies  in  the 
canal,  and  on  account  of  its  inertia  lags  behind,  while  the  relative  shifting  of  the 
water  along  the  wall  of  the  canal  might  be  felt  by  the  ciliae  of  the  nerves  of  the 
ampullae.  On  the  other  hand,  if  the  turning  continues,  the  ring  of  water  itself 
will  be  gradually  set  in  revolution  by  its  friction  against  the  wall  of  the  canal, 
and  will  continue  to  move,  even  when  the  turning  of  the  head  suddenly  ceases. 
This  causes  the  illusive  sensation  of  a  revolution  in  the  contrary  direction,  in  the 
well-known  form  of  giddiness.  Injuries  to  the  semicircular  canals  without  injuries 
to  the  brain  produce  the  most  remarkable  disturbances  of  equilibrium  in  the  lower  H 
animals.  Electrical  discharges  through  the  ear  and  cold  water  squirted  into  the 
ear  of  a  person  with  a  perforated  drumskin,  produce  the  most  violent  giddiness. 
Under  these  circumstances  these  parts  of  the  ear  can  no  longer  with  any  probability 
be  considered  as  belonging  to  the  sense  of  hearing.  Moreover  impulses  of  the 
stirrup  against  the  water  of  the  labyrinth  adjoining  the  oval  window  are  in  reality 
ill  adapted  for  prodiicing  streams  through  the  semicircular  canals. 

On  the  other  hand  the  experiments  of  Koenig  with  short  sounding  rods,  and 
those  of  Preyer  with  Appunn's  tuning-forks,  have  established  the  fact  that  very 
high  tones  with  from  4000  to  40,000  vibrations  in  a  second  can  be  heard,  but  that 
for  these  the  sensation  of  interval  is  extremely  deficient.  Even  intervals  of  a  Fifth 
or  an  Octave  in  the  highest  positions  are  only  doubtfully  recognised  and  are  often 
wrongly  appreciated  by  practised  musicians.  Even  the  major  Third  c' — e'  [4096  : 
5120  vibrations]  was  at  one  time  heard  as  a  Second,  at  another  as  a  Fourth  or  a 
Fifth ;  and  at  still  greater  heights  even  Octaves  and  Fifths  were  confused.  H 

If  we  maintain  the  hypothesis,  that  every  nervous  fibre  hears  in  its  own  peculiar 
jiitch,  we  should  have  to  conclude  that  the  vibrating  parts  of  the  ear  which  convey 
these  sensations  of  the  highest  tones  to  the  ear,  are  much  less  sharply  defined  in  their 
capabilities  of  resonance,  than  those  for  deeper  tones.  This  means  that  they  lose  any 
motion  excited  in  them  comparatively  soon,  and  are  also  comparatively  more  easily 
Ttrought  into  the  state  of  motion  necessary  for  sensation.  This  last  assumption 
must  be  made,  because  for  parts  which  are  so  strongly  damped,  the  possibility  of 
adding  together  many  separate  impulses  is  very  limited,  and  the  construction  of  the 
auditory  ciliae  in  the  little  liags  of  the  otoliths  seems  to  me  more  suited  for  this 
purpose  than  that  of  the  shortest  fibres  of  the  basilar  membrane.  If  this  hypo- 
thesis is  confirmed  we  should  have  to  regard  the  auditory  ciliae  as  the  bearers  of 
squeaking,  hissing,  chirping,  crackling  sensations  of  sound,  and  to  consider  their 
reaction  as  diftering  only  in  degree  from  that  of  the  cochlear  fibres,  f 

*  Ffluecjer,    Archir.     fur    Fhysiolof/ic,    vol.  t  [See   App.  XX.  sect.  L.  art.  5.— Trans-. 

xiii.  '  Iiifor.] 


PAKT   II. 

ON  THE  INTERRUPTIONS  OF  HARMONY 


COMBINATIONAL  TONES  AND  BEATS, 
CONSONANCE  AND  DISSONANCE. 


CHAPTER  VII. 

COMBINATIONAL    TONES. 

In  the  first  part  of  this  book  we  had  to  enunciate  and  constantly  apply  the  pro- 
position that  oscillatory  motions  of  the  air  and  other  elastic  bodies,  produced  by 
several  sources  of  sound  acting  simultaneously,  are  always  the  exact  sum  of  the 
individual  motions  producible  by  each  source  separately.  This  law  is  of  extreme 
importance  in  the  theory  of  sound,   because  it  reduces  the  consideration  of  com- 

^  pound  cases  to  those  of  simple  ones.  But  it  must  be  observed  that  this  law  holds 
strictly  only  in  the  case  where  the  vibrations  in  all  parts  of  the  mass  of  air  and  of 
the  sonorovis  elastic  bodies  are  of  infinitesimally  small  dimensions ;  that  is  to  say, 
only  when  the  alterations  of  density  of  the  elastic  bodies  are  so  small  that  they 
may  be  disregarded  in  comparison  with  the  whole  density  of  the  same  body ;  and 
in  the  same  way,  only  when  the  displacements  of  the  vibrating  particles  vanish  as 
compared  with  the  dimensions  of  the  Avhole  elastic  body.  Now  certainly  in  all 
practical  applications  of  this  law  to  sonorous  bodies,  the  vibrations  are  always 
verj/  small,  and  near  enough  to  being  infinitesimalli/  small  for  this  law  to  hold 
with  great  exactness  even  for  the  real  sonorous  vibrations  of  musical  tones,  and  by 
far  the  greater  part  of  their  phenomena  can  be  deduced  from  that  law  in  con- 
formity with  observation.  Still,  however,  there  are  certain  phenomena  which 
result  from  the  fact  that  this  law  does  not  hold  with  perfect  exactness  for  vibra- 
tions of  elastic  bodies,  which,  though  almost  always  very  small,  are  far  from  being 
infinitesliHally  small.-f     One  of  these  phenomena,  with  which  we  are  here  interested 

5f  is  the  occurrence  of  Comhinational  Tones,  which  were  first  discovered  in  1745  by 
Sorge,!  a  German  organist,  and  were  afterwards  generally  known,  although  their 
pitch  was  often  wrongly  assigned,  through  the  Italian  violinist  Tartini  (1754),  from 
whom  they  are  often  called  Tartini's  tones.^ 

These  tones   are  heard   whenever  two  musical    tones  of   ditterent    iiitches   are 

*  [So  much  attention   has   recently  been  holtz's  views  before  taking  up  the  Appendix, 

paid  to  the  whole  subject  of  this  second  part  — I'ranslator.'] 

— Combinational    Tones    and   Beats — mostly  +  Helmholtz,  on   '  Combinational   Tones,' 

since   the   publication    of    the    4th    German  in    Poggendorf's   Annalea,   vol.    xcix.   p.  497. 

edition,  that  I  have  thought  it  advisable  to  Moiwtsbcrkhlc  of  the  Berlin  Academy,  ^Nlay  2-2, 

give  a  brief  account  of  the  investigations  of  1856.     From  this  last  au  extract  is  given  in 

Koenig,  Bosauquet,  and  Preyer  in  App.  XX.  Appendix  XII. 

sect.   L.,  and  merely  add  a  few  footnotes  to  \    J'oryrinach      musikalischer      Compos itioib 

refer  the  reader  to  them  where  they  especially  (Antechamber  of  musical  composition), 
relate  to  the  statements  in  the  text.     But  the  §  [In  England  they  have  hence  been  often 

reader  should  study  the  text  of  this  second  called  by  Tartini's  name,  icrzi  siwui,  or  third 

part,   so  as  to  be  familiar  with  Prof.  Helm-  sounds,  resulting  from  the  combination  of  two. 


CHAP.  VII.  COMBINATIONAL  TONES.  153 

sounded  together,  loudly  and  continuously.  The  pitch  of  a  combinational  tone 
is  generally  different  from  that  of  either  of  the  generating  tones,  or  of  their 
harmonic  upper  partials.  In  experiments,  the  combinational  are  readily  distin- 
guished from  the  upper  partial  tones,  by  not  being  heard  when  only  one  generating 
tone  is  sounded,  and  by  appearing  simultaneously  with  the  second  tone.  Combi- 
national tones  are  of  two  kinds.  The  first  class,  discovered  by  Sorge  and  Tartini, 
I  have  termed  differential  tones,  because  their  pitch  number  is  the  difference  of 
the  pitch  numbers  of  the  generating  tones.  The  second  class  of  siimmationnl 
tones,  having  their  pitch  number  equal  to  the  sum  of  the  pitch  numbers  of  the 
generating  tones,  were  discovered  by  myself. 

On  investigating  the  combinational  tones  of  two  compound  unisical  tones,  we 
find  that  both  the  primary  and  the  upper  partial  tones  may  give  rise  to  both  dif- 
ferential and  summational  tones.  In  such  cases  the  number  of  combinational 
tones  is  very  great.  P"-t  it  must  be  observed  that  generally  the  differential  are  % 
stronger  than  the  summational  tones,  and  that  the  stronger  generating  simple 
tones  also  produce  the  stronger  combinational  tones.  The  combinational  tones, 
::i  k.u,  increase  in  a  much  greater  ratio  than  the  generating  tones,  and  diminish 
also  more  rapidly.  Now  since  in  musical  compound  tones  the  prime  generally  pre- 
dominates over  the  partials,  the  differential  tones  of  the  two  primes  are  generally 
heard  more  loudly  than  all  the  rest,  and  were  consequently  first  discovered.  They 
are  most  easily  heard  when  the  two  generating  tones  are  less  than  an  octave  apart, 
because  in  that  case  the  differential  is  deeper  than  either  of  the  two  generating 
tones.  To  hear  it  at  first,  choose  two  tones  which  can  be  held  with  great  force  for 
some  time,  and  form  a  justly  intoned  harmonic  interval.  First  sound  the  low 
tone  and  then  the  high  one.  On  properly  directing  attention,  a  weaker  low  tone 
will  be  heard  at  the  moment  that  the  higher  note  is  struck ;  this  is  the  required 
combinational  tone.*  For  pai'ticular  instruments,  as  the  harmonium,  the  com- 
binational tones  can  be  made  more  audible  by  properly  tuned  resonators.  In  this  *\ 
case  the  tones  are  generated  in  the  air  contained  within  the  instrument.  But  in 
other  cases,  where  chey  are  generated  solely  within  the  ear,  the  resonators  are  of 
little  or  no  use. 

A  commoner  English  name  is  rjrave  harmonics,  The  differential  tones  are  well  heard  on  the 
which  is  inapplicable,  as  they  are  not  neces-  English  concertina,  for  the  same  reason  as  on 
sarily  graver  than  both  of  the  generating  tones.  the  harmonium.  High  notes  forming  Semi- 
Prof.  Tyndall  calls  them  rcsidtant  tones.  I  tones  tell  well.  It  is  convenient  to  choose 
prefer  retaining  the  Latin  expression,  first  in-  close  dissonant  intervals  for  first  examples  in 
troduced,  as  Prof.  Preyer  informs  us  {Akusti-  order  to  dissipate  the  old  notion  that  the 
sdie  UntcrsurlnuKicn,  ^.  11), hy  Q.\5.  A.yieth.  'grave  harmonic'  is  necessarily  the  '  true 
(d.  1836  in  Dessau)  in  Gilbert's  Annulen  der  fundamental  bass'  of  the  'cliord'.  It  is  very 
Physih  1805,  vol.  xxi.  p.  265,  but  only  for  the  easy  when  playing  two  high  generating  notes, 
tones  here  distinguished  as  differential,  and  as  ij'"  and  ,</"';|f  or  the  last  and  a'",  to  hear  at 
afterwards  used  by  Scheibler  and  Prof.  Helm-  the  same  time  the  rattle  of  the  beats  (see  next 
holtz.  I  shall,  however,  use  'combinational  chapter)  and  the  deep  combinational  tones 
tones  ■  to  express  all  the  additional  tones  which  about  FJ^  and  (?,|j,  much  resembling  a  thrash- 
are  heard  when  two  notes  are  sounded  at  the  ing  machine  two  'or  three  fields  off.  The  beats  ^ 
same  time.— Translator. ^  and  the  differentials  have  the  same  frequency 
*  [I  have  found  that  combinational  tones  (note  p.  lid).  See  infra,  App.  XX.  sect.  L.  art. 
can  be  made  quite  audible  to  a  hundred  people  5,  /.  The  experiment  can  also  be  made  with 
at  once,  by  means  of  two  flageolet  fifes  or  //"  c"  and  h'"^  h"  on  any  harmonium.  And  if 
whistles,  blown  as  strongly  as  possible.  I  all  three  notes  h"\),  b",  c'"  are  held  down  to- 
chooseveiyclose  dissonant  intervals  because  the  gether,  the  ear  can  perceive  the  two  sets  of 
great  depth  of  the  low  tone  is  much  more  strik-  beats  of  the  upper  notes  as  sharp  high  rattles, 
ing,  being  very  far  below  anything  that  can  be  and  the  beats  of  the  two  combinational  tones, 
touched  by  the  instrument  itself.  Thus  </'"  about  the  pitch  of  C,  which  have  altogether  a 
being  sounded  loudly  on  one  fife  by  an  assis-  different  character  and  frequency.  On  the 
taut,  I  give  /'""£,  when  a  deep  note  is  instantly  Harmonical,  notes  //'  c'"  should  beat  66,  notes 
heard  which,  if  the  interval  were  pure,  would  b"\f  b"  should  beat  39-6,  and  notes  'b"\}  ¥"\f 
be  (/,  and  is  sufficiently  near  to  (j  to  be  recog-  should  beat  26-4  in  a  second,  and  these  should 
nised  as  extremely  deep.  As  a  second  experi-  be  the  pitches  of  their  combinational  notes ; 
ment,  the  r/""  being  held  as  before,  I  give  first  the  two  first  should  therefore  beat  26-4  times 
/""ti  and  then  c""  in  succession.  If  the  inter-  in  a  second,  and  the  two  last  13-2  times  in  a 
vals  were  pure  the  combinational  tones  would  second.  But  the  tone  26-4  is  so  difficult  to 
jump  from  q  to  c",  and  in  reality,  the  jump  is  hear  that  the  beats  are  not  distinct. — Trmu- 
very  nearly  the  same  and  quite  appreciable.  lator.] 


154 


COMBINATIONAL  TONES. 


The   following  table  gives    the   first  difterential   tones  of  the   usual   harmonic 
intervals : — 


Ratio  of  the 
vibrational 
numbers 


Difference  of 

The  combinational  tone  is  deeper  than   i 

the  same 

a  Unison 

an  Octave 

a  Twelfth 

Two  Octaves 

Two  Octaves  and  a  major  Third 

2 

a  Fifth 

3 

a  major  Sixth 

'  Octave 
I  Fifth 

Fourth     . 
1  Major  Third 
I  Minor  Third 
I  Major  Sixtli 

Minor  Sixth 


1  :  2 

2:  3 

3  :  4 

4  :  5 

5  :  6 
3  :  5 
5  :  8 


or  in  ordinary  musical  notation,  the  generating  tones  being  written  as  minims  and 
H  the  difterential  tones  as  crotchets — 


Octave.       Fifth.       Fourth. 


:\Iajor 
Third. 


Minor 
Third. 


Major 
Sixth. 


Minor 
Sixth. 


e^ 


1^1 


5: 


When  the  ear  has  learned  to  hear  the  combinational  tones  of  pure  intervals 
and  sustained  tones,  it  will  be  able  to  hear  them  from  inharmonic  intervals  and  in 
the  rapidly  fading  notes  of  a  pianoforte.  The  combinational  tones  from  inhar- 
^  monic  intervals  are  more  difficult  to  hear,  because  these  intervals  beat  more  or  less 
strongly,  as  we  shall  have  to  explain  hereafter.  The  combinational  tones  arising 
from  such  as  fade  rapidly,  for  example  those  of  the  pianoforte,  are  not  strong 
enough  to  be  heard  except  at  the  first  instant,  and  die  oft"  sooner  than  the  gene- 
rating tones.  Combinational  tones  are  also  in  general  easier  to  hear  from  the  simple 
tones  of  tuning-forks  and  stopped  organ  pipes  than  from  compound  tones  where  a 
number  of  other  secondary  tones  are  also  present.  These  compound  tones,  as  has 
been  already  said,  also  generate  a  number  of  difterential  tones  by  their  harmonic 
upper  partials,  and  these  easily  distract  attention  from  the  difterential  tones  of  the 
primes.  Combinational  tones  of  this  kind,  arising  from  the  upper  partials,  are 
frequently  heard  from  the  violin  and  harmonium. 

Example. — Take  the  major  Third  c'e',  ratio  of  pitch  numbers  4  :  5.  First  difference  1,  that 
is  C.  The  first  harmonic  upper  partial  of  c'  is  c",  relative  pitch  number  8.  Ratio  of  this  and 
c',  5  :  8,  difference  3,  that  is  g.  The  first  upper  partial  of  r'  is  c",  relative  pitch  number  10 ; 
^  ratio  for  this  and  c',  4  :  10,  difference  6,  that  is  ;/.  Then  again  c"  e"  have  ratio  8  :  10,  difference 
2,  that  is  '■.  Hence,  taking  only  the  first  upper  partials  we  have  the  series  of  combinational 
tones  1,  3,  6,  2  or  C,  ij,  r/',  c.     Of  these  the  tone  3,  or  g,  is  often  easily  perceived. 

These  multiple  combinational  tones  cannot  in  general  be  distinctly  heard,  except 
when  the  generating  compound  tones  contain  audible  harmonic  upper  partials. 
Yet  we  cannot  assert  that  the  combinational  tones  are  absent,  where  such  partials 
are  absent  ;  but  in  that  case  they  are  so  weak  that  the  ear  does  not  readily  recognise 
them  beside  the  loud  generating  tones  and  first  difterential.  In  the  first  place 
theory  leads  us  to  conclude  that  they  do  exist  in  a  weak  state,  and  in  the  next 
place  the  beats  of  impiu-e  intervals,  to  be  discussed  presently,  also  establish  their 
existence.  In  this  case  we  may,  as  Hallstroem  suggests,*  consider  the  multiple 
combinational  tones  to  arise  thus :  the  first  difterential  tone,  or  cambinational  tone 
of  the  first  order,  by  combination  with  the  generating  tones  themselves,  produce 
other  difterential  tones,   or  comhinational   tones  of  the  second  order ;   these   again 

*  Poggendorff's  Annakn,  vol.  xxiv.  p.  438. 


CHAP.  VII. 


COMBINATIONAL  TONES. 


155 


produce   new   ones   with    the  generators    and    dittorentials   of    the  first   order,  and 
so  on. 

Example. — Take  two  simple  tones  c'  tand  c',  ratio  4:  5,  dift'ereuce  1,  diftereiitial  tone  of  tlio 
first  order  C.  This  with  tlio  generators  gives  the  ratios  1 :  4  and  1 :  5,  differences  3  and  4, 
differential  tones  of  tlie  second  order  g,  and  c'  once  more.  The  new  tone  3,  gives  with  the 
generators  the  ratios  3  :  4  and  3  :  5,  differences  1  and  2,  giving  the  differential  tones  of  the  third 
order  C  and  e,  and  the  same  tone  3  gives  with  the  differential  of  the  first  order  1,  the  ratio  1 :  3, 
difference  2,  and  hence  as  a  differential  of  the  fourth  order  c  once  more  and  so  on.  The  dif- 
ferential tones  of  different  orders  which  coincide  when  the  interval  is  perfect,  as  it  is  supposed  to 
be  in  this  example,  no  longer  exactly  coincide  when  the  generating  interval  is  not  pure;  and 
consequently  such  beats  are  heard,  as  would  result  from  the  presence  of  these  tones.  IMore  on 
this  hereafter. 

The  differential  tones  of  different  orders  for  different   intervals  arc  given  in  the 
following  notes,  where  the  generators  are  minims,  the  combinational  tones  of  the  1! 
first  order  crotchets,  of  the  second  quavers,  and  so  on.     The  same  tones  also  occur 
with  compound  generators  as  combinational  tones  of  their  upper  partials.* 


Fourth. 


Major  Third. 


Minor  Third. 


Major  Sixth 


Minor  Sixth 


BI 


:^^:^g=^_Tr=::tzzzt=^=i=rpi= 

1 — /._. 1 H A 1 


The  series  are  broken  off  as  soon  as  the  last  order  of  differentials  furnishes  no 
fresh  tones.  In  general  these  examples  shew  that  the  complete  series  of  harmonic 
partial  tones  I,  2,  3,  4,  5,  kc,  up  to  the  generators  themselves,t  is  produced. 

The  second  kind  of  combinational  tones,  which  I  have  distinguished  as  summa- 
tional, is  generally  nuich  weaker  in  sound  than  the  first,  and  is  only  to  be  heard 


*  [These  examples  are  best  calculated  by 
iving  to  the  notes  in  the  example  the  numbers 
^presenting  the  harmonics  on  p.  22r.".     Thus 

Octave,  notes  4 :  S.     Diff.  8-4  =  4. 

Fifth,  notes  4  :  6.     Diff.  6-4  =  2. 
2nd  order,  4-2  =  2,  6-2  =  4. 

2. 


Fourth,  notes  6 :  8.     Diff.  8  -  6  = 
2nd  order,  8-2  =  6,6-2  =  4. 
3rd  order,  6-4  =  2,6-2  =  4. 


]\Iajor  Third,  notes  4  :  5. 

2nd.  4-1  =  3,  5-1  =  4. 

3rd.    4-3  =  1,  5-3  =  2. 

4th.    4-2  =  2,  4  -1  =  3. 
Minor  Third,  notes  5  :  G. 

2nd.  5-1  =  4,  6-1  =  5. 

3rd.    5-4  =  1,6-4  =  2. 

4th.    4-1  =  3,  6-2  =  4. 

5th.   6-4  =  2,6-3  =  3. 


Diff.  5-4  =  1. 


Diff.  0-5  =  1. 


Major  Sixth,  notes  6 :  10.     Diff.  10-6  =  4.    ^ 
2nd.  10-4  =  6,  6-4  =  2. 
3rd.    10-2  =  8,6-2  =  4. 
4th.    6-4  =  2. 

Minor  Sixth,  notes  5:8.  Diff.  8  -  5  =  3. 
2nd.  5-3  =  2,8-3  =  5. 
3rd.  5-2  =  3,  8-2  =  6. 
4th.  3-2  =  1,  5-3  =  2. 
5th.  5-1  =  4,  8-1=7. 
6th.    8-7  =  5-4=1,4-2  =  2,8-4  =  4. 

The  existence  of  these  differential  tones  of 
higher  orders  cannot  be  considered  as  com- 
pletely established. — Translator.] 

t  [See  App.  XX.  sect.  L.  art.  7,  for  the 
influence  of  such  a  series  on  the  consonance  of 
simple  tones.  It  is  not  to  be  supposed  that  ah 
these  tones  are  audible.  ]\Ir.  Bosanquet  derives 
them  direct  from  the  generators,  see  App.  XX. 
sect.  L.  art.  5,  a. — Translator. \ 


156 


COMBINATIONAL  TONES. 


with  decent  ease  under  peculiarly  favourable  circumstances  on  the  harmonium  and 
polyphonic  siren.  Scarcely  any  but  the  first  summational  tone  can  be  perceived, 
having  a  vibrational  number  equal  to  the  sum  of  those  of  the  generators.  Of  course 
STinmiational  tones  may  also  arise  from  the  harmonic  upper  partials.  Since  their 
vibrational  number  is  always  equal  to  the  sum  of  the  other  two,  they  are  always 
higher  in  pitch  than  either  of  the  two  generators.  The  following  notes  will  shew 
their  nature  for  the  simple  intervals:  — 


a: 


^^=: 


-j — r^*~7T~ 


fc. — ^- 


— r 

Octave. 


Fifth. 


m 


2  +  4  2  +  3 

=  6.  =5. 

In  relation  to  music  I 


Fourth. 

3  +  4 

=  7. 

will   here 


:^: 


Major 
Sixth. 
3  +  5 


Major 
Third. 

4  +  5 
=  9. 


Minor 
Third. 
5  +  6 
=  11. 


Minor 

Sixth.* 
5  +  8 
=  13. 


remark  at  once  that  many  cf  these  summa- 
tional tones  form  extremely  inharmonic  intervals  with  the  generators.  Were  they 
not  generally  so  weak  on  most  instruments,  they  would  give  rise  to  intoler- 
able dissonances.  In  reality,  the  major  and  minor  Third,  and  the  minor  Sixth, 
sound  very  badly  indeed  on  the  polyphonic  siren,  where  all  combinational  tones 
are  remarkably  loud,  whereas  the  Octave,  Fifth,  and  major  Sixth  are  very  beautifid. 
Even  the  Fourth  on  this  siren  has  only  the  effect  of  a  tolerably  harmonious  chord 
of  the  minor  Seventh. 
^  It  was  formerly  believed  that  the  combinational  tones  were  purely  subjective, 
and  were  produced  in  the  ear  itself.t  Differential  tones  alone  were  known,  and  these 
were  connected  with  the  beats  which  usually  result  from  the  simultaneous  sounding 
of  two  tones  of  nearly  the  same  pitch,  a  phenomenon  to  be  considered  in  the  follow- 
ing chapters.  It  was  believed  that  when  these  beats  occurred  with  sufficient 
rapidity,  the  individual  increments  of  loudness  might  produce  the  sensation  of  a 
new  tone,  just  as  numerous  ordinary  impulses  of  the  air  would,  and  that  the 
frequency  of  such  a  tone  would  be  equal  to  the  frequency  of  the  beats.  But  this 
supposition,  in  the  first  place,  does  not  explain  the  origin  of  summational  tones, 
being  confined  to  the  differentials  ;  secondly,  it  may  be  proved  that  under  certain 
conditions  the  combinational  tones  exist  objectively,  independently  of  the  ear 
which  would  have  had  to  gather  the  beats  into  a  new  tone  ;  and  thirdly,  this 
supjjosition  cannot  be  reconciled  with  the  law  confirmed  by  all  other  experiments, 
that  the  only  tones  which  the  ear  hears,  correspond  to  pendular  vibrations,  of  the 
fair.+ 

And  in  reality  a  diiferent  cause  for  the  origin  of  combinational  tones  can  be 
established,  which  has  already  been  mentioned  in  general  terms  (p.  1 52c).  When- 
ever the  vibrations  of  the  air  or  of  other  elastic  bodies  which  are  set  in  motion  at 
the  same  time  by  two  generating  simple  tones,  are  so  powerful  that  they  can  no  longer 
be  considered  infinitely  small,  mathematical  theory  shows  that  vibrations  of  the 
air  must  arise  which  have  the  same  frequency  as  the  combinational  tones.  § 

Particular  instruments  give  very  powerful  combinational  tones.     The  condition 

objections,  and  for  other  objections,  see  App. 
XX.  sect.  L.  art.  5,  b,  c. — Trans/ator.] 

§  [The  tones  supposed  to  arise  from  beats, 
and' the  differential  tones  thus  generated,  are 
essentially  distinct,  having  sometimes  the  same 
but  frequently  different  pitch  numbers.  See 
App.  XX.  sect.  L.  art.  3,  d. — Translatm-.A 


*  [The  notation  of  the  last  5  bars  has  been 
altered  to  agree  with  the  diagram  of  harmonics 
of  Con  p.  12c. -Translator.'] 

+  [The  result  of  Mr.  Bosanquet's  and  Prof. 
Preyer's  quite  recent  experiments  is  to  shew 
that  they  are  so.  See  App.  XX.  sect.  L.  art.  4, 
b,  c. —  Translafm:] 

J  [For   Prof.    Preyer's   remarks   on   these 


CHAP.  VII.  COMBINATIONAL  TONES.  157 

foi-  their  generation  is  that  the  same  mass  of  air  should  be  violently  agitated  by 
two  simple  tones  simultaneously.  This  takes  place  most  powerfully  in  the  poly- 
phonic siren,*  in  which  the  same  rotating  disc  contains  two  or  more  series  of 
holes  which  are  blown  upon  simultaneously  from  the  same  windchest.  The  air 
of  the  windchest  is  condensed  whenever  the  holes  are  closed  ;  on  the  holes  being 
o])ened,  a  large  quantity  of  air  escapes,  and  the  pressure  is  considerably  diminished. 
Consequently  the  air  in  the  windchest,  and  partly  even  that  in  the  bellows,  as 
can  be  easil}^  felt,  comes  into  violent  vibration.  If  two  rows  of  holes  are  blown, 
vibrations  arise  in  the  air  of  the  windchest  con-espouding  to  both  tones,  and  each 
row  of  oj)enings  gives  vent  not  to  a  stream  of  air  uniformly  supplied,  but  to  a 
stream  of  air  already  set  in  vibration  by  the  other  tone.  Under  these  circumstances 
the  combinational  tones  are  extremely  powerful,  almost  as  powerful,  indeed,  as  the 
generators.  Their  objective  existence  in  the  mass  of  air  can  be  proved  by  vibra- 
ting membranes  tuned  to  be  in  unison  with  the  combinational  tones.  Such  U 
membranes  are  set  in  sympathetic  vibration  immediately  upon  both  generating 
tones  being  sounded  simultaneously,  but  remain  at  rest  if  only  one  or  the  other  of 
them  is  sounded.  Indeed,  in  this  case  the  summational  tones  are  so  powerful 
that  they  make  all  chords  extremely  unpleasant  which  contain  Thirds  or  minor 
Sixths.  Instead  of  membranes  it  is  more  convenient  to  use  the  resonators  already 
recommended  for  investigating  harmonic  upper  partial  tones.  Resonators  are 
also  unable  to  reinforce  a  tone  when  no  pendular  vibrations  actually  exist  in  the 
air ;  they  have  no  effect  on  a  tone  which  exists  only  in  auditoiy  sensation,  and 
hence  they  can  be  used  to  discover  whether  a  combinational  tone  is  objectively 
present.  They  are  much  more  sensitive  than  membranes,  and  are  well  adapted 
for  the  clear  recognition  of  very  weak  objective  tones. 

The  conditions  in  the  harmonium  are  similar  to  those  in  the  siren.  Here,  too, 
there  is  a  common  windchest,  and  when  two  keys  are  pressed  down,  we  have  two 
openings  which  are  closed  and  opened  rhythmically  by  the  tongues.  In  this  case  *H 
also  the  air  in  the  common  receptacle  is  violently  agitated  by  both  tones,  and  air 
is  blown  through  each  opening  which  has  been  ah'eady  set  in  vibration  by  the 
other  tongue.  Hence  in  this  instrument  also  the  combinational  tones  are  objectively 
present,  and  comparatively  very  distinct,  but  they  are  far  from  being  as  powerful 
as  on  the  siren,  probably  because  the  windchest  is  very  much  larger  in  proportion 
to  the  openings,  and  hence  the  air  which  escaj^es  during  the  short  opening  of  an 
exit  by  the  oscillating  tongue  cannot  be  sufficient  to  diminish  the  pressure  sensibly. 
In  the  harmonium  also  the  combinational  tones  are  very  clearly  reinforced  by 
resonators  tuned  to  be  in  unison  with  them,  especially  the  first  and  second  dif- 
ferential and  the  first  summational  tone.f  Nevertheless  I  have  convinced  myself,  by 
particular  experiments,  that  even  in  this  instrument  the  greater  part  of  the  force 
of  the  combinational  tone  is  generated  in  the  ear  itself.  I  arranged  the  portvents 
in  the  instrument  so  that  one  of  the  two  generators  was  supplied  with  air  by  the 
bellows  moved  below  by  the  foot,  and  the  second  generator  was  blown  by  theH 
reserve  bellows,  which  was  first  pumped  full  and  then  cut  off  by  drawing  out  the 
so-called  expression-stop,  and  I  then  found  that  the  combinational  tones  were  not 
much  weaker  than  for  the  usual  arrangement.  But  the  objective  portion  which 
the  resonators  reinforce  was  much  weaker.  The  noted  examples  given  above 
(pp.  154-5-6)  will  easily  enable  any  one  to  find  the  digitals  which  must  be 
pressed  down  in  order  to  produce  a  combinational  tone  in  unison  with  a  given 
resonator. 

On  the  other  hand,  when  the  places  in  which  the  two  tones  are  struck  are 
entirely  separate  and  have  no  mechanical  connection,  as,  for  example,  if  they  come 
from  tw^o  singers,  two  separate  wind  instruments,  or  two  violins,  the  reinforcement 

*  A  detailed  description  of  this  instrument  apparent  reinforcement  by  a  resonator  arose 

will  be  given  in  the  next  chapter.  from  imperfect  blocking  of   both   ears  when 

t  [The  experiments  of  Bosanquet,  App.  XX.  using  it.  See  also  p.  iSd',  note.— Translator.] 
sect.  L.  art.  4,  b,  render  it  probable  that  this 


158  COMBINATIONAL  TONES.  part  ii. 

of  the  combinational  tones  by  resonators  is  small  and  du))ions.  Here,  then,  there 
does  not  exist  in  the  air  any  clearly  sensible  pendular  vibration  corresponding  to 
the  combinational  tone,  and  we  must  conclude  that  such  tones,  which  are  often 
powerfully  audible,  are  really  produced  in  the  ear  itself.  But  analogously  to  the 
former  cases  we  are  justified  in  assuming  in  this  case  also  that  the  external  vibra- 
ting parts  of  the  ear,  the  drumskin  and  auditory  ossicles,  are  really  set  in  a  suffi- 
ciently powerful  combined  vibration  to  generate  combinational  tones,  so  that  the 
vibrations  which  correspond  to  combinational  tones  may  really  exist  objectively  in 
the  parts  of  the  ear  without  existing  objectively  in  the  external  air.  A  slight  rein- 
forcement of  the  combinational  tone  in  this  case  by  the  corresponding  resonator 
may,  therefore,  arise  from  the  drumskin  of  the  ear  communicating  to  the  air  in  the 
resonator  those  particular  vibrations  which  con-espond  to  the  combinational  tone.* 

Now  it  so  happens  that  in  the  construction  of  the  external  parts  of  the  ear  for 
^  conducting  sound,  there  are  certain  conditions  which  are  peculiarly  favourable  for 
the  generation  of  combinational  tones.  First  we  have  the  unsymmetrical  form  of 
the  drumskin  itself.  Its  radial  fibres,  which  are  externally  convex,  undergo  a  much 
greater  alteration  of  tension  when  they  make  an  oscillation  of  moderate  amplitude 
towards  the  inside,  than  when  the  oscillation  takes  place  towards  the  outside. 
For  this  purpose  it  is  only  necessary  that  the  amj)litude  of  the  oscillation  shovild 
not  be  too  small  a  fraction  of  the  minute  depth  of  the  arc  made  by  these  radial 
fibres.  Under  these  circumstances  deviations  from  the  simple  superposition  of 
vibrations  arise  for  very  tnuch  smaller  amplitudes  than  is  the  case  when  the  vibra- 
ting body  is  symmetrically  constructed  on  both  sides. f 

But  a  more  important  circumstance,  as  it  seems  to  me,  when  the  tones  are 
powerful,  is  the  loose  formation  of  the  joint  between  the  hammer  and  anvil  (p.  1336). 
If  the  handle  of  the  hammer  is  driven  inwards  by  the  drumskin,  the  anvil  and 
stirrup  must  follow  the  motion  unconditionally.  But  that  is  not  the  case  for  the 
^  subseqiient  outward  motion  of  the  handle  of  the  hammer,  dnring  which  the  teeth 
of  the  two  ossicles  need  not  catch  each  other.  In  this  case  the  ossicles  may  dick. 
Now  I  seem  to  hear  this  clicking  in  my  own  ear  whenever  a  very  strong  and  deep 
tone  is  brought  to  bear  upon  it,  even  when,  for  example,  it  is  produced  by  a  tuning- 
fork  held  between  the  fingers,  in  which  there  is  certainly  nothing  that  can  make 
any  click  at  all. 

This  peculiar  feeling  of  mechanical  tingling  in  the  ear  had  long  ago  struck  me 
when  two  clear  and  powerful  soprano  voices  executed  passages  in  Thirds,  in  which 
3ase  the  combinational  tone  comes  out  very  distinctly.  If  the  phases  of  the  two 
tones  are  so  related  that  after  every  fourth  oscillation  of  the  deeper  and  every  fifth 
of  the  higher  tone,  there  ensues  a  considerable  outward  displacement  of  the  drum- 
skin, sufficient  to  cause  a  momentary  loosening  in  the  joint  between  the  hammer 
and  anvil,  a  series  of  blows  will  be  generated  between  the  two  bones,  which  would 
be  absent  if  the  connection  were  firm  and  the  oscillation  regular,  and  these  blows 
^  taken  together  would  exactly  generate  the  first  differential  tone  of  the  interval  of 
a  major  Third.     Similarly  for  other  intervals. 

It  must  also  be  remarked  that  the  same  peculiarities  in  the  construction  of  a 
sonorous  body  which  makes  it  suitable  for  allowing  combinational  tones  to  be  heard 
when  it  is  excited  by  two  waves  of  different  pitch,  must  also  cause  a  single  simple 
tone  to  excite  in  it  vibrations  corresponding  to  its  harmonic  upper  partials ;  the 
effect  being  the  same  as  if  this  tone  then  formed  svmmiational  tones  with  itself. 

This  result  ensues  because  a  simple  periodical  force,  corresponding  to  pendular 
vibrations,  cannot  excite  similar  pendular  vibrations  in  the  elastic  body  on  which 
it  acts,  unless  the  elastic  forces  called  into  action  by  the  displacements  of  the  ex- 

*  [See    latter    half    of    Appendix    XVI. —  are  proportional  to  the  first  power  of  the  am- 

Translator.']  plitude,   whereas   for  symmetrical  ones  they 

t  See   my  paper   on  combinational  tones  are  proportional  to  only  the  second  power  of 

already  cited,  and  Appendix  XII.    For  unsym-  this  magnitude,  which  is  very  small  in  both 

metrical   vibrating    bodies    the    disturbances  cases. 


€HAPS.  VII.  VIII.  INTERFERENCE  OF  SOT'ND.  159 

cited  body  from  its  position  of  equilibrium,  are  proportional  to  these  dis])Uicements 
themselves.  This  is  always  the  case  so  long  as  these  displacements  are  infinitesimal. 
But  if  the  amplitude  of  the  oscillations  is  great  enough  to  cause  a  sensible  devia- 
tion from  this  proportionality,  then  the  vibrations  of  the  exciting  tone  are  increased 
by  others  which  con-espond  to  its  harmonic  upper  partial  tones.  That  such  har- 
monic upper  partials  are  occasionally  heard  when  tuning-forks  are  strongly  ex- 
cited, has  been  already  mentioned  (p.  54(/).  I  have  lately  repeated  these  experi- 
ments with  forks  of  a  very  low  pitch.  With  such  a  fork  of  64  vib.  I  could,  by 
means  of  proper  resonators,  hear  up  to  the  fifth  partial.  But  then  the  amplitude 
of  the  vibrations  was  almost  a  centimetre  [-3937  inch].  When  a  sharp-edged 
body,  such  as  the  prong  of  a  tuning-fork,  makes  vibrations  of  such  a  length, 
vortical  motions,  differing  sensibly  from  the  law  of  simple  vibrations,  must  arise 
in  the  surrounding  air.  On  the  other  hand,  as  the  sound  of  the  fork  fades,  these 
upper  partials  vanish  long  before  their  prime,  which  is  itself  only  very  weakly  H 
audible.  This  agrees  with  our  hypothesis  that  these  partials  arise  from  disturb- 
ances depending  on  the  size  of  the  amplitude. 

Herr  R.  Koenig,*  with  a  series  of  forks  having  sliding  weights  by  which  the  pitch 
might  be  gradually  altered,  and  provided  also  with  boxes  giving  a  good  resonance 
and  possessing  powerful  tones,  has  investigated  beats  and  combinational  tones,  and 
found  that  those  combinational  tones  were  most  prominent  w^hich  answered  to  the 
difference  of  oae  of  the  tones  from  the  partial  tone  of  the  other  which  was  nearest 
to  it  in  pitch ;  and  in  this  research  partial  tones  as  high  as  the  eighth  were  effec- 
tive (at  least  in  the  number  of  beats ).t  He  has  unfortunately  not  stated  how  far 
the  corresponding  upper  partials  were  separately  recognised  by  resonators.  J 

Since  the  human  ear  easily  produces  combinational  tones,  for  which  the  prin- 
cipal causes  lying  in  the  construction  of  that  organ  have  just  been  assigned,  it 
must  also  form  upper  partials  for  powerful  simple  tones,  as  is  the  case  for  tuning- 
forks  and  the  masses  of  air  which  they  excite  in  the  observations  described.  Hence  ^ 
we  cannot  easily  have  the  sensation  of  a  jwtverful  simple  tone,  without  having  also 
the  sensation  of  its  harmonic  upper  partials.§ 

The  importance  of  combinational  tones  in  the  construction  of  chords  will  appear 
hereafter.  We  have,  however,  first  to  investigate  a  second  phenomenon  of  the 
simultaneous  sounding  of  two  tones,  the  so-called  beats. 


CHAPTER  VIII. 

ON    THK    BEATS    OF    SIMPLE    TONES. 


Wb  now  pass  to  the  consideration  of  other  phenomena  accompanying  the  simul-  ^ 
taneous  sounding  of  two  simple  tones,  in  which,  as  before,  the  motions  of  the  air 
and  of  the  other  co-operating  elastic  bodies  without  and  within  the  ear  may  be  con- 
ceived as  an  undisturbed  coexistence  of  two  systems  of  vibrations  corresponding  to 
the  two  tones,  but  where  the  auditory  sensation  no  longer  corresponds  to  the  sum 
of  the  two  sensations  which  the  tones  would  excite  singly.  Beats,  wiiich  have 
now  to  be  considered,  are  essentially  distinguished  from  combinational  tones  as 
follows  : — In  combinational  tones  the  composition  of  vibrations  in  the  elastic 
vibrating  bodies  which  are  either  within  or  without  the  ear,  undergoes  cei'tain  dis- 
turbances, although  the  ear  resolves  the  motion  which  is  finally  conducted  to  it, 

*  Poggendorff's  Annul.,  vol.  clvii.  pp.  177-  sect.  L. — Translator.] 
236.  I  [Koenig   states   that  no   upper   partials 

t  [Even  with  this  parenthetical  correction,  could  be  heard.     See  Appendix  XX.  sect.  L. 

the  above  is  calculated  to  give  an  inadequate  art.  2,  a. — Translator.] 

impression  of  the  results  of  Koenig's  paper,  §  [See  App.  XX.  sect.  L.  art.  1,  ii. — Trans- 

which  is  more  fully  described  in  Appendix  XX.  lator.] 


160 


INTERFERENCE  OF  SOUND. 


into  a  series  of  simple  tones,  according  to  the  usual  law.  In  beats,  on  the 
contrary,  the  objective  motions  of  the  elastic  bodies  follow  the  simple  law ;  but 
the  composition  of  the  sensations  is  disturbed.  As  long  as  several  simple  tones  of 
a  sufficiently  different  pitch  enter  the  ear  together,  the  sensation  due  to  each 
remains  undisturbed  in  the  ear,  probably  because  entirely  different  bundles  of 
nerve  libres  are  affected.  But  tones  of  the  same,  or  of  nearly  the  same  pitch, 
which  therefore  affect  the  same  nerve  fibres,  do  not  produce  a  sensation  which  is 
the  sum  of  the  two  they  would  have  separately  excited,  but  new  and  peculiar 
phenomena  arise  which  we  term  interference,  when  caused  by  two  perfectly  equal 
simple  tones,  and  heats  when  due  to  two  nearly  equal  simple  tones. 

We  will  begin  with  interference.  Suppose  that  a  point  in  the  air  or  ear 
is  set  in  motion  by  some  sonorous  force,  and  that  its  motion  is  represented  by 
the  curve  1,  fig.  53.     Let 

Uthe    second     motion    be  ^'"-  ^^■ 

precisely  the  same  at  the 
same  time  and  be  repre- 
sented by  the  cui-ve  2,  so 
that  the  crests  of  2  fall 
on  the  crests  of  1,  and 
also  the  troughs  of  2  on 
the  troughs  of  1 .  If  both 
motions  proceed  at  once, 
the  whole  motion  will  be 
their  sum,  represented  by 
3,  a  curve  of  the  same 
kind  but  with  crests  twice  as  high  and  troughs  twice  as  deep  as  those  of  either  of 
the    others.       Since  the  intensity  of   sound    is    proportional    to  the  square  of  the 

H  amplitude,  we  have  consequently  a  tone  not  of  twice  but  of  four  times  the  loudness 
of  either  of  the  others. 

But  now  suppose  the  vibrations  of  the  second  motion  to  be  displaced  by 
half  the  periodic  time.  The  curves  to  be  added  will  stand  under  one  another,  as 
4  and  5  in  fig.  54,  and 
when  we  come  to  add 
to  them,  the  heights  of 
the  second  curve  will  be 
still  the  same  as  those 
of  the  first,  but,  being 
always  in  the  contrary 
direction,  the  two  will 
mutually  destroy  each 
other,     giving    as    their 

^ sum  the  straight  line  6,  or  no  vibration  at  all.  In. this  case  the  crests  of  4  are 
added  to  the  troughs  of  5,  and  conversely,  so  that  the  crests  fill  up  the  troughs, 
and  crests  and  troughs  mutually  annihilate  each  other.  The  intensity  of  sound 
also  becomes  nothing,  and  when  motions  are  thus  cancelled  within  the  ear,  sensa- 
tion also  ceases ;  and  although  each  single  motion  acting  alone  would  excite  the 
corresponding  auditory  sensation,  when  both  act  together  there  is  no  auditory 
sensation  at  all.  ()ne  sound  in  this  case  completel}'  cancels  what  appears  to  be 
an  equal  sound.  This  seems  extraordinarily  paradoxical  to  ordinarj-  contempla- 
tion because  our  uatui-al  conscioiisness  apprehends  sound,  not  as  the  motion  of 
particles  of  the  air,  but  as  something  really  existing  and  analogous  to  the  sensation 
of  sound.  Now  as  the  sensation  of  a  simple  tone  of  the  same  pitch  shows  no  oppo- 
sitions of  positive  and  negative,  it  naturally  appears  impossible  for  one  positive 
sensation  to  cancel  another.  Bnt  the  really  cancelling  things  in  such  a  case  are 
the  vibrational  impulses  which  the  two  sources  of  soimd  exert  on  the  ear.  When 
it  so  happens  that  the  vibrational  impulses  due  to  one  source  constantly  coincide 


CHAP.  VIII.  INTERFERENCE  OF  SOUND.  161 

with  opposite  ones  due  to  the  other,  and  exactly  countcil)alance  each  other,  no 
motion  can  possibly  ensue  in  the  ear,  and  hence  the  auditory  nerve  can  experience 
no  sensation. 

The  following  are  some  instances  of  sound  cancelling  sound  : — 
Put  two  perfectly  similar  stopped  organ  jMpes  tuned  to  the  same  pitch  close 
beside  each  other  on  the  same  portvent.  Each  one  blown  separately  gives  a 
powerful  tone  ;  but  when  they  are  blown  together,  the  motion  of  the  air  in  the 
two  pipes  takes  place  in  such  a  manner  that  as  the  air  streams  out  of  one  it  streams 
into  the  other,  and  hence  an  observer  at  a  distance  hears  no  tone,  but  at  most  the 
rushing  of  the  air.  On  bringing  the  fibre  of  a  feather  near  to  the  lips  of  the 
pipes,  this  fibre  will  vibrate  in  the  same  way  as  if  each  pipe  were  blown  separately. 
Also  if  a  tube  be  conducted  from  the  ear  to  the  mouth  of  one  of  the  pipes,  the 
tone  of  that  pipe  is  heard  so  much  more  powerfully  that  it  cannot  be  entirely 
destroyed  by  the  tone  of  the  other.*  ^ 

Every  tuning-fork  also  exhibits  phenomena  of  interference,  because  the  prongs 
move  in  opposite  directions.  On  striking  a  tuning-fork  and  slowly  revolving  it 
about  its  longitudinal  axis  close  to  the  ear,  it  will  be  found  that  there  are  four 
positions  in  which  the  tone  is  heard  clearly;  and  four  intermediate  positions  in 
which  it  is  inaudil)le.  The  four  positions  of  strong  sound  are  those  in  which 
either  one  of  the  prongs,  or  one  of  the  side  surfaces  of  the  fork,  is  turned  towards 
the  ear.  The  positions  of  no  sound  lie  between  the  former,  almost  in  planes 
which  make  an  angle  of  45°  with  the  surfaces  of  the  prongs,  and  pass  through 
the  axis  of  the  fork.  If  in  fig.  55,  a  and  b  are  the  ends  of  the  fork  seen  from 
above,  c,  d,  e,  f  will  be  the  four  places  of  strong  sound,  and  the  dotted  lines 
Pi^  gg  the  four  places  of  silence.     The  arrows  under  a 

and  b  shew  the  nuitual  motion  of  the  two  prongs. 
/      Hence  while  the  prong  a  gives  the  air  about  c  an  im- 
/        pulse  in  the  direction  c  a,  the  prong  b  gives  it  an  U 
opposite  one.     Both  impulses  only  partially  cancel 
one  another  at  c,  because  a  acts  more  powerfully 
than  b.    But  the  dotted  lines  shew  the  places  where 
the  opposite  impulses  from  a  and  b  are  equally 
*^  strong,  and  consequently  com|)letely  cancel  each 

~^  "*\  other.     If  the  ear  be  brought  into  one  of  these 

f         \  places  of  silence  and  a  narrow  tube  be  slipped  over 

one  of  the  prongs  a  or  b,  taking  care  not  to  touch  it, 
the  sound  will  be  immediately  augmented,  because 
/'  \^         the  influence  of  the  covered  prong  is  almost  entirely 

destroyed,  and  the  uncovered  prong  therefore  acts 
alone  and  imdisturbed.j 
A  double  siren  which  I  have  had  constructed  is  very  convenient  for  the  demon- 
stration of  these  relations.  J  Fig.  56  (p.  162)  is  a  perspective  view  of  this  instru-H 
ment.  It  is  composed  of  two  of  Dove's  polyphonic  sirens,  of  the  kind  previously 
mentioned,  p.  1.3a;  a„  and  ai  are  the  two  windchests,  c„  and  c,  the  discs  attached 
to  a  common  axis,  on  which  a  screw  is  introduced  at  k,  to  drive  a  counting 
apparatus  which  can  be  introduced,  as  described  on  p.  \1b.  The  upper  box  a, 
can  be  turned  round  its  axis,  by  means  of  a  toothed  Avheel,  in  which  works  a 
smaller  wheel  e  provided  with  the  driving  handle  d  §.  The  axis  of  the  box  a, 
round  which  it  turns,  is  a  prolongation  of  the  upper  pipe  gi,  which  conducts 
the  wind.     On  each  of  the  two  discs  of  the  siren  are  four  rows  of  holes,  which 

*  [If  a  screen  of  any  sort,  as  the  hand,  be  resonance  chamber,  the  alternation  of  sound 

interposed  between  the  moutlis  of  the  pipes,  and  silence,  &c.,  can  be  made  audible  to  many 

the   tone  is   immediately  restored,  and   then  persons  at  once.— rra^^sto/or.] 
generally   remains   even   if   the   hand  be    re-  +  Constructed  by  the  mechanician  Sauer- 

moved.— rra?is/«to/-.]  wald  in  Berlin. 

+  [If  instead  of  bringing  the  tuning-fork  to  §  [Three   turns  of   the  handle   cause   one 

the  ear,  it  be  slowly  turned  before  a  proper  turn  of  the  box  round  its  axis.— 7'm/i.sZa<or.] 

M 


162 


INTEKFEKENCK  OF  SOUND. 


can  be  either  blown  separately  or  together  in  any  combination  at  pleasure,  and  at  i 
are  the  studs  for  opening  and  closing  the  series  of  holes  by  a  peculiar  arrange- 
ment.* The  lower  disc  has  four  rows  of  8,  10,  12,  18  holes,  the  upper  of  9,  12, 
15,   16.      Hence  if  we  call  the  tone  of  8  holes  c,  the  lower  disc  gives  the  tones  c,  e^ 


g,  d'  and  the   upper  (/,   g,  h,  c .     "We  are  therefore  able  to  produce  the  following 
intervals : — 

1 .  Unison  :  gg  on  the  two  discs  simultaneously. 

2.  Octaves  :  c  c'  and  d  d'  on  the  two. 


*  Described  in  Appendix  XIII. 


CHAP.  VIII.  INTERFERENCE  OF  SOUND.  16;? 

3.  Fifths  :  c  g  and  //  d'  either  on  the  lower  disc  alone  or  on  both  discs  to"-ether. 

4.  Fourths  :  d  (j  and  g  c  on  the  upper  disc  alone  or  on  both  together. 

5.  Major  Third  :  c  e  on  the  lower  alone,  and  g  b  on  the  upper  alone,  or  q  h  on 
both  together. 

6.  Minor  Third  :  e  g  o\\  the  lower,  or  on  both  together ;  h  d'  on  both  together. 

7.  Whole  Tone  [major  Tone] :  c  d  and  c  d'  on  both  together  [the  minor  Tone 
is  produced  by  d  and  e  on  both  together]. 

8.  Semitone  [diatonic  Semitone]  :  b  c  on  the  upper. 

When  both  tones  are  produced  from  the  same  disc  the  objective  combinational 
tones  are  very  powerful,  as  has  been  already  remarked,  p.  157«.  But  if  the  tones 
are  produced  from  different  discs,  the  combinational  tones  are  weak.  In  the  latter 
case  (and  this  is  the  chief  point  of  interest  to  us  at  present),  the  two  tones  can 
be  made  to  act  together  with  any  desired  difference  of  phase.  This  is  effected  by 
altering  the  position  of  the  upper  box.  ^ 

We  have  first  to  investigate  the  phenomena  as  they  occur  in  the  unison  g  g. 
The  effect  of  the  interference  of  the  two  tones  in  this  case  is  complicated  by  the 
fact  that  the  siren  produces  compound  and  not  simple  tones  and  that  the  in- 
terference of  individual  partial  tones  is  independent  of  that  of  the  prime  tone 
and  of  one  another.  In  order  to  damp  the  upper  partial  tones  in  the  siren  by 
means  of  a  resonance  chamber,  I  caused  cylindrical  boxes  of  brass  to  be  made, 
of  which  the  back  halves  are  shewn  at  hi  h,  and  h^  ho  fig.  56,  opposite.  These 
boxes  are  each  made  in  two  sections,  so  that  they  can  be  removed,  and  be  again 
attached  to  the  windchest  by  means  of  screws.  When  the  tone  of  the  siren 
approaches  the  prime  tone  of  these  boxes,  its  quality  becomes  full,  strong  and  soft, 
like  a  fine  tone  on  the  French  horn ;  otherwise  the  siren  has  rather  a  piercing  tone. 
At  the  same  time  we  use  a  small  quantity  of  air,  but  considerable  pressure.  The 
circumstances  are  of  the  same  nature  as  when  a  tongue  is  applied  to  a  resonance 
chamber  of  the  same  pitch.  Used  in  this  way  the  siren  is  very  well  adapted  for  ^ 
experiments  on  interference. 

If  the  boxes  are  so  placed  that  the  puffs  of  air  follow  at  exactly  equal  intervals 
from  both  discs,  similar  phases  of  the  prime  tone  and  of  all  partials  coincide,  and 
all  are  reinforced. 

If  the  handle  is  turned  round  half  a  right  angle,  the  upper  box  is  turned  round 
I  of  a  right  angle,  or  -^  of  the  circumference,  that  is  half  the  distance  between 
the  holes  in  the  series  of  12  holes  which  is  in  action  for  ^.  Hence  the  difference 
in  the  phase  of  the  two  primes  is  half  the  vibrational  period,  the  puffs  of  air  in 
one  box  occur  exactly  in  the  middle  between  those  of  the  other,  and  the  two 
](rime  tones  mutually  destroy  each  other.  But  vmder  these  circumstances  the 
difterence  of  phase  in  the  upper  Octave  is  precisely  the  whole  of  the  vibrational 
period ;  that  is,  they  reinforce  each  other,  and  similarly  all  the  evenly  numbered 
harmonic  upper  partials  reinforce  each  other  in  the  same  position,  and  the  unevenly 
numbered  ones  destroy  each  other.  Hence  in  the  new  position  the  tone  is  weaker,  1] 
because  deprived  of  several  of  its  partials ;  but  it  does  not  entirely  cease ;  it  rather 
jumps  up  an  Octave.  If  we  further  turn  the  handle  through  another  half  a  right 
angle  so  that  the  box  is  turned  through  a  whole  right  angle,  the  pufts  of  the  two 
discs  again  agree  completely,  and  the  tones  reinforce  one  another.  Hence  in  a 
complete  revolution  of  the  handle  there  are  four  positions  where  the  whole  tone  of 
the  siren  appears  reinforced,  and  four  intermediate  positions  where  the  prime  tone 
and  all  uneven  upper  partials  vanish,  and  consequently  the  Octave  occurs  in  a 
weaker  form  accompanied  by  the  evenly  numbered  upper  partials.  If  we  attend  to 
the  first  upper  partial,  which  is  the  Octave  of  the  prime,  by  listening  to  it  through 
a  proper  resonator,  we  find  that  it  vanishes  after  turning  through  a  quarter  of  a 
right  angle,  and  is  reinforced  after  turning  through  half  a  right  angle,  and  hence 
for  every  complete  revolution  of  the  handle  it  vanishes  8  times,  and  is  reinforced 
8  times.  The  third  partial  (or  second  upper  partial),  the  Twelfth  of  the  prime 
tone,  vanishes  in  the  same  time  12  times,  the  fourth  partial  16  times,  and  so  on. 

m2 


164  •  ORIGIN  OF  BEATS.  part  ii. 

Other  compound  tones  behave  like  those  of  the  siren.  When  two  tones  of  the 
same  pitch  are  sounded  together  having  differences  of  phase  corresponding  to  half 
the  periodic  time,  the  tone  does  not  vanish,  but  jumps  up  an  Octave.  When,  for 
example,  two  open  organ  pipes,  or  two  reed  pipes  of  the  same  construction  and 
pitch,  are  placed  beside  each  other  on  the  same  portvent,  their  vibrations  generally 
accommodate  themselves  in  such  a  manner  that  the  stream  of  air  enters  first  one 
and  then  the  other  alternately  •  and  while  the  tone  of  stopped  pipes,  which  have 
only  unevenly  numbered  partials,  is  then  almost  entirely  destroyed,  the  tone  of  the 
open  pipes  and  reed  pipes  falls  into  the  upper  Octave.  This  is  the  reason  why  no 
reinforcement  of  tone  can  be  effected  on  an  organ  or  harmonium  by  combining 
tongues  or  pipes  of  the  same  kind  [on  the  same  portventj. 

So  far  we  have  combined  tones  of  precisely  the  same  pitch ;  now  let  us  inquire 
what  happens  when  the  tones  have  slightly  different  pitch.  The  double  siren 
^  just  described  is  also  well  fitted  for  explaining  this  case,  for  we  can  slightly  alter 
the  pitch  of  the  upper  tone  by  slowly  revolving  the  upper  box  by  means  of  the 
handle,  the  tone  becoming  flatter  when  the  direction  of  revolution  is  the  same  as 
that  of  the  disc,  and  sharper  when  it  is  opposite  to  the  same.  The  vibrational 
period  of  a  tone  of  the  siren  is  equal  to  the  time  required  for  a  hole  in  the  rotating 
disc  to  pass  from  one  hole  in  the  windbox  to  the  next.  If,  through  the  rotation  of 
the  box,  the  hole  of  the  box  advances  to  meet  the  hole  of  the  disc,  the  two  holes 
come  into  coincidence  sooner  than  if  the  box  were  at  rest :  and  hence  the  vibra- 
tional period  is  shorter,  and  the  tone  sharper.  The  converse  takes  place  when  the 
revolution  is  in  the  opposite  direction.  These  alterations  of  pitch  ai'e  easily  heard 
when  the  box  is  revolved  rather  quickly.  Now  produce  the  tones  of  tw^elve  holes 
•  on  both  discs.  These  will  be  in  absolute  unison  as  long  as  the  upper  box  is  at 
rest.  The  two  tones  constantly  reinforce  or  enfeeble  each  other  according  to  the 
position  of  the  upper  box.  But  on  setting  the  upper  box  in  motion,  the  pitch  of 
H  the  upper  tone  is  altered,  while  that  of  the  lower  tone,  Avhich  has  an  immovable 
windbox,  is  unchanged.  Hence  we  have  now  two  tones  of  slightly  different  pitch 
sounding  together.  And  w^e  hear  the  so-called  h'^ats  of  the  tones,  that  is,  the 
intensity  of  the  tone  will  be  alternately  greater  and  less  in  regular  succession.*  The 
arrangement  of  our  siren  makes  the  reason  of  this  readily  intelligible.  The 
revolution  of  the  upper  box  brings  it  alternately  in  positions  which  as  wc  have 
seen  correspond  to  stronger  and  weaker  tones.  When  the  handle  has  been  turned 
through  a  right  angle,  the  windbox  passes  from  a  position  of  loudness  through  a 
position  of  weakness  to  a  position  of  strength  again.  Consequently  every  complete 
revolution  of  the  handle  gives  us  four  beats,  whatever  be  the  rate  of  revolution  of 
the  discs,  and  hence  however  low  or  high  the  tone  may  be.  If  we  stop  the  box  at 
the  moment  of  maximum  loudness,  we  continue  to  hear  the  loud  tone  ;  if  at  a 
moment  of  minimum  force,  we  continue  to  hear  the  weak  tone. 

The  mechanism  of  the  instrument  also  explains  the  connection  between  the 
*\  number  of  beats  and  the  difference  of  the  pitch.  It  is  easily  seen  that  the  number 
of  the  puffs  is  increased  by  one  for  every  quarter  revolution  of  the  handle.  But 
every  such  quarter  revolution  corresponds  to  one  beat.  Hence  the  number  of  heats 
in  a  given  time  is  equal  to  the  difference  of  the  numbers  of  vibrations  executed  by 
the  tivo  tones  in  the  same  time.  This  is  the  general  law  which  determines  the 
number  of  beats,  for  all  kinds  of  tones.  This  law  results  immediately  from  the 
construction  of  the  siren;  in  other  instruments  it  can  only  be  verified  by  very 
accurate  and  laborious  measurements  of  the  numbers  of  vibrations. 

The  process  is  shewai  graphically  in  fig.  57.  Here  c  c  represents  the  series  of 
puffs  belonging  to  one  tone,  and  d  d  those  belonging  to  the  other.  The  distance 
for  0  c  is  divided  into  18  parts,  the  same  distance  is  divided  into  20  parts  for  d  d.    At 

*  [The  German  word  Schvwbmig,  which  '  beat '.  But  it  is  not  usual  to  make  the  dis- 
might  be  rendered  'fluctuation,'  implies  this:  tinction  in  English,  where  the  whole  pheno- 
The  loudest  portion  only  is  called  the  Stoss,  or       menon  is  called  beats.  —  Translator.] 


CHAP.  VIII.  ORIGIN  OF  BEATS.  165 

1,  3,  5,  both  piifts  concur,  and  the  tone  is  reinforced.  At  2  and  4  they  are  inter- 
mediate and  mutually  enfeeble  each  other.  The  number  of  beats  for  the  whole 
distance  is  2,  because  the  difterence  of  the  numbers  of  parts,  each  of  which  cor- 
respond to  a  vibration,  is  also  2. 

The  intensity  of  tone   varies  ;  swelling  from  a  minimum  to  a  maximimi,  and 
lessening    from    the  maximum    to   the    minimum.      It  is  the  places  of   maximum 


Ot  3  r  5  C 

I,      I       l'     :'     l'    f  I     'l      'l      'l       I       l'      i'      l'     ■'    .    '   i    'l     ',      'l       I 

intensity  which  are  properly  called  beaU,  and  these  are   separated  by  more  or  less 
distinct  pauses. 

Beats  are  easily  produced  on  all  musical  instruments,  by  striking  two  notes  of  H 
nearly  the  same  pitch.  They  are  heard  best  from  the  simple  tones  of  tuning-forks 
or  stopped  organ  pipes,  because  here  the  tone  really  vanishes  in  the  pauses.  A 
little  fluctuation  in  the  pitch  of  the  beating  tone  may  then  be  remarked.*  For  the 
compound  tones  of  other  instruments  the  upper  partial  tones  are  heard  in  the 
pavises,  and  hence  the  tone  jumps  up  an  Octave,  as  in  the  case  of  interference 
already  described.  If  we  have  two  tuning-forks  of  exactly  the  same  pitch,  it  is 
only  necessary  to  stick  a  little  wax  on  to  the  end  of  one,  to  strike  both,  and  bring 
them  near  the  same  ear  or  to  the  surface  of  a  table,  or  sounding  board.  To  make 
two  stopped  pipes  beat,  it  is  only  necessary  to  bring  a  finger  slowly  near  to  the  lip 
of  one,  and  thus  flatten  it.  The  beats  of  compound  tones  are  heard  by  striking 
any  note  on  a  pianoforte  out  of  tune  when  the  two  strings  belonging  to  the  same 
note  are  no  longer  in  unison  ;  or  if  the  piano  is  in  tune  it  is  sufficient  to  attach  a 
piece  of  wax,  about  the  size  of  a  pea,  to  one  of  the  strings.  This  puts  them  suffi- 
ciently out  of  tune.  More  attention,  however,  is  required  for  compound  tones  H 
because  the  enfeeblement  of  the  tone  is  not  so  striking.  The  beat  in  this  case 
resembles  a  fluctuation  in  pitch  and  quality.  This  is  very  striking  on  the  siren 
according  as  the  brass  resonance  cylinders  (h^  h„  and  hi  hj  of  fig.  56,  p.  162)  are 
attached  or  not.  These  make  the  prime  tone  relatively  strong.  Hence  when  beats 
are  produced  by  turning  a  handle,  the  decrease  and  increase  of  loudness  in  the  tone 
is  very  striking.  On  removing  the  resonance  cylinders,  the  upper  partial  tones 
are  relatively  powerful,  and  since  the  ear  is  very  uncertain  when  comparing  the 
loudness  of  tones  of  different  pitch,  the  alteration  of  force  during  the  beats  is 
much  less  striking  than  that  of  pitch  and  quality  of  tone. 

On  listening  to  the  upper  partials  of  compound  tones  which  beat,  it  will  be 
found  that  these  beat  also,  and  that  for  each  beat  of  the  prime  tone  there  are  two 
of  the  second  partial,  three  of  the  third  and  so  on.  Hence  when  the  upper  partials 
are  strong,  it  is  easy  to  make  a  mistake  in  counting  the  beats,  especially  when  the 
beats  of  the  primes  are  very  slow,  so  that  they  occur  at  intervals  of  a  second  or  two.  H 
It  is  then  necessary  to  pay  great  attention  to  the  pitch  of  the  beats  counted,  and 
sometimes  to  apply  a  resonator. 

It  is  possible  to  render  beats  visible  by  setting  a  suitable  elastic  body  into 
sympathetic  vibration  with  them.  Beats  can  then  occur  only  when  the  two 
exciting  tones  lie  near  enough  to  the  prime  tone  of  the  sympathetic  body  for  the 
latter  to  be  set  into  sensible  sympathetic  vibration  by  both  the  tones  used.  This 
is  most  easily  done  with  a  thin  string  which  is  stretched  on  a  sounding  board 
on  which  have  been  placed  two  tuning-forks,  both  of  very  nearly  the  same  pitch 
as  the  string.  On  observing  the  vibrations  of  the  string  through  a  microscope, 
or  attaching  a  fibril  of  a  goosefeather  to  the  string  which  will  make  the  same 
excursions  on  a  magnified  scale,  the  string  will  be  clearly  seen  to  make  sympathetic 

*  See  the  explanation  of  this  phenomenon  French  translator  of  this  work,]  in  Appen- 
which  was  given  me  by  Mons.  G.  Gueroult  [the       dix  XIV. 


166  ORIGIN  OF  BEATS.  part  ii. 

vibrations  with  alternately  large  and  small  excursions,  according  as  the  tone  of  the 
two  forks  is  at  its  maximum  or  minimum. 

The  same  effect  can  be  obtained  from  the  sympathetic  vibration  of  a  stretched 
membrane.     Fig,  58  is  the  copy  of  a  drawing  made  by  a  vibrating  membrane  of 

Fio.    58. 


this  sort,  used  in  the  phonautograph  of  Messrs.  Scott  it  Koenig,  of  Paris.  The  mem- 
brane of  this  instrument,  which  resembles  the  drumskin  of  the  ear,  carries  a  small 
stiff  style,  which  draws  the  vibrations  of  the  membrane  upon  a  rotating  cylinder. 
In  the  ])resent  case  the  membrane  was  set  in  motion  by  two  organ  jjipes,  that  beat. 

H  The  undulating  line,  of  which  only  a  pai't  is  here  given,  shews  that  times  of  strong 
vibration  have  alternated  with  times  of  almost  entire  rest.  In  this  case,  then,  the 
beats  are  also  sympathetically  executed  by  the  membrane.  Similar  drawings 
again  have  been  made  by  Dr.  Politzer,  who  attached  the  Avriting  style  to  the 
auditory  bone  (the  columella)  of  a  duck,  and  then  produced  a  beating  tone  by 
means  of  two  organ  pipes.  This  experiment  shewed  that  even  the  auditory  bones 
follow  the  beats  of  two  tones.* 

Generally  this  must  always  be  the  case  when  the  pitches  of  the  two  tones 
struck  differ  so  little  from  each  other  and  from  that  of  the  proper  tone  of  the  sym- 
pathetic body,  that  the  latter  can  be  put  into  sensible  vibration  by  both  tones  at 
once.  Sympathetic  bodies  which  do  not  damp  readily,  such  as  tuning-forks, 
consequently  require  two  exciting  tones  Avhich  diifer  extraordinarily  little  in  pitch, 
in  order  to  shew  visible  beats,  and  the  beats  must  therefore  be  very  slow.  For 
bodies  readily  damped,  as   membranes,   strings,   &c.,  the  difference  of  the  exciting 

^  tones  ma}-  be  greater,  and  consequently  the  beats  may  succeed  each  other  more 
rapidly. 

This  holds  also  for  the  elastic  terminal  formations  of  the  auditory  nerve  fibres. 
Just  as  we  have  seen  that  there  may  be  visible  beats  of  the  auditor}^  ossicles,  Corti's 
arches  may  also  be  made  to  beat  by  two  tones  sufficiently  near  in  pitch  to  set  the 
same  Corti's  arches  in  sympathetic  vibration  at  the  same  time.  If  then,  as  we 
have  previously  supposed,  the  intensity  of  auditory  sensation  in  the  nerve  fibres 
involved  increases  and  decreases  with  the  intensity  of  the  elastic  vibrations,  the 
strength  of  the  sensation  must  also  increase  and  diminish  in  the  same  degree  as  the 
vibrations  of  the  corresponding  elastic  appendages  of  the  nerves.  In  this  case  also 
the  motion  of  Corti's  arches  must  still  be  considered  as  compounded  of  the  motions 
which  the  two  tones  would  have  produced  if  they  had  acted  separately.  According 
as  these  motions  are  directed  in  the  same  or  in  opposite  directions  they  will  rein- 
force or  enfeeble  each  other  by  (algebraical)  addition.     It  is  not  till  these  motions 

H  excite  sensation  in  the  nerves  that  any  deviation  occurs  from  the  law  that  each  of 
the  two  tones  and  each  of  the  two  sensations  of  tones  subsist  side  b\-  side  without 
disturbance. 

We  now  come  to  a  part  of  the  investigation  which  is  very  important  for  the 
theory  of  musical  consonance,  and  has  also  unfortunately  been  little  regarded  by 
acousticians.  The  question  is  :  what  becomes  of  the  beats  when  they  grow  faster 
and  faster  1  and  to  what  extent  may  their  number  increase  without  the  ear  being 
unable  to  perceive  them?  Most  acousticians  were  probably  inclined  to  agree  with 
the  hypothesis  of  Thomas  Young,  that  when  the  beats  became  very  (piick  they 
gradually  passed  over  into  a  combinational  tone  (the  first  differential).  Young 
imagined  that  the  pulses  of  tone  which  ensue  during  beats,  might  have  the  same 

*  The  beats  of  two  tones  are  also  clearly  tones.    Even  without  using  the  rotating  mirror 

shewn  by  the  vibrating  flame  described  at  the  for  observing  the  flames,  we  can  easily  recog- 

end  of  Appendix  II.     The  flame  must  be  con-  nise  the  alterations  in  the  shape  of  the  flame 

nected  with  a  resonator  having  a  pitch  sufii-  which    takes    place   isochronously   with    the 

ciently   near  to   those  of  the  two  generating  audible  beats. 


CHAP.  VIII.  LIMITS  OF  THE  FREQUENCY  OF  BFATS.  167 

effect  on  the  ear  as  elementary  pulses  of  air  (in  the  siren,  for  exaaiple),  and  that 
just  as  30  puffs  in  a  second  through  a  siren  would  produce  the  sensation  of  a  deep 
tone,  so  would  30  beats  in  a  second  resulting  from  any  two  higher  tones  produce 
the  same  sensation  of  a  deep  tone.  Certainly  this  view  is  well  supported  by  the 
fact  that  the  vibrational  number  of  the  first  and  strongest  combinational  tone  is 
actually  the  number  of  beats  produced  by  the  two  tones  in  a  second.  It  is,  however, 
of  much  importance  to  remember  that  there  are  other  combinational  tones  (my 
summational  tones),  which  will  not  agree  with  this  hypothesis  in  any  respect,'* 
but  on  the  other  hand  are  readily  deduced  from  the  theory  of  combinational  tones 
which  I  have  proposed  (in  Appendix  XII.).  It  is  moreover  an  objection  to  Young's 
theor}^,  that  in  many  cases  the  combinational  tones  exist  externally  to  the  ear,  and 
are  able  to  set  properly  tuned  membranes  or  resonators  into  sympathetic  vibra- 
tion,t  because  this  could  not  possibly  be  the  case,  if  the  combinational  tones  were 
nothing  but  the  series  of  beats  with  undisturbed  superposition  of  the  two  waves.  ^ 
For  the  mechanical  theory  of  sympathetic  vibration  shews  that  a  motion  of  the 
air  compounded  of  two  simple  vibrations  of  different  periodic  times,  is  capable  of 
putting  such  bodies  only  into  sympathetic  vibration  as  have  a  proper  tone  corre- 
sponding to  one  of  the  two  given  tones,  provided  no  conditions  intervene  by  which 
the  simple  superposition  of  two  wave  systems  might  be  disturbed  ;  and  the  nature 
of  such  a  disturbance  was  investigated  in  the  last  chapter.:):  Hence  we  may 
consider  combinational  tones  as  an  accessory  phenomenon,  by  which,  however,  the 
course  of  the  two  primary  wave  systems  and  of  their  beats  is  not  essentially 
interrupted. 

Against  the  old  opinion  we  may  also  adduce  the  testimony  of  our  senses,  which 
teaches  us  that  a  much  greater  number  of  beats  than  30  ii^  a  second  can  be 
distinctly  heard.  To  obtain  this  result  we  must  pass  gradually  from  the  slower  to 
the  more  rapid  beats,  taking  care  that  the  tones  chosen  for  beating  are  not  too  far 
apart  from  each  other  in  the  scale,  because  audible  beats  are  not  produced  unless  H 
the  tones  are  so  near  to  each  other  in  the  scale  that  they  can  both  make  the  same 
elastic  appendages  of  the  nerves  vibrate  sympathetically.  §  The  number  of  beats, 
however,  can  be  increased  without  increasing  the  interval  between  the  tones,  if 
both  tones  are  taken  in  the  higher  octaves. 

The  observations  are  best  begun  by  producing  two  simple  tones  of  the  same 
pitch,  say  from  the  once-accented  octave  by  means  of  tuning-forks  or  stopped  organ 
pipes,  and  slowly  altering  the  pitch  of  one.  This  is  effected  by  sticking  more  and 
more  wax  on  one  of  the  forks ;  or  more  and  more  covering  the  mouth  of  one  of 
the  pipes.  Stopped  organ  pipes  are  also  generally  provided  with  a  movable  plug 
or  lid  at  the  stopped  end,  in  order  to  time  them  ;  by  pulling  this  out  we  flatten,  by 
pushing  it  in  we  sharpen  the  tone.** 

When  a  slight  difference  in  pitch  has  been  thus  produced,  the  beats  are  heard 
at  first  as  long  drawn  out  fluctuations  alternately  swelling  and  vanishing.     Slow 
beats  of  this  kind  are  by  no  means  disagreeable  to  the  ear.     In  executing  music  ^ 
containing  long  sustained  chords,  they  may  even  produce  a  solemn  effect,  or  else 
give  a  more  lively,  tremulous  or  agitating  expression.     Hence  we   find  in   modern 

*  [Prof.  Preyer  shews,  App.  XX.  sect.  L.  stration  of  the  following  facts,  is  made  with 

art.  4,  d,  that  summational  tones,  as  suggested  two  '  pitch  pipes,'  each  consisting  of  an  exten- 

by  Appunn,  may  be  considered  as  differential  sible  stopped  pipe,  which  has  the  compass  of 

tonesof  the  second  order,  if  such  are  admitted.  the  once-accented  octave  and  is  blown  as  a 

—  Translator.']  whistle,  the  two  being  connected  by  a  bent  tube 

t  [After  the  experiments  of   Prof.  Preyer  with  a  single  mouthpiece.    By  carefully  adjust- 

and  INIr.  Bosanquet,  App.  XX.  sect.  L.  art.  4,  ing  the  lengths  of  the  pipes,  I  was  first  able  to 

this  must  be  considered  as  due  to  some  error  produce  complete  destruction  of  the  tone  by 

of  observation. — Translator.']  interference,  the  sound  returning  inunediately 

X  [See  Bosanquet's  theory  of  '  transforma-  wlien  the  mouth  of  one  whistle  was  stopped  by 

tion'  in  App.  XX.  sect.  L.  art.  5,  «.— '/'m/;*-  the  finger.    Then  on  gradually  lengthening  one 

lator.]  of  the  pipes  the  beats  began  to  be  heard  slowly, 

§  [  Koenig  knows  no  such  limitation.     See  and  increased  in  rapidity.      The  tone  being 

App.  XX.  sect.  L.  art.  3. — Translator.]  nearly  simple   the   beats    are   well    heard. — 

**  [A  cheap  apparatus,  useful  for  demon-  Translator.] 


168  LIMITS  OF  THE  FREQUENCY  OF  BEATS.  pakt  ii. 

organs  and  harmoniums,  a  stop  with  two  pipes  or  tongues,  adjusted  to  beat.  This 
imitates  the  trembling  of  the  human  voice  and  of  violins  which,  appropriately  in- 
troduced in  isolated  passages,  may  certainly  be  very  expressive  and  effective,  but 
applied  continuously,  as  is  unfortunately  too  common,  is  a  detestable  malpractice. 

The  ear  easily  follows  slow  beats  of  not  more  than  4  to  6  in  a  second.  The 
hearer  has  time  to  apprehend  all  their  separate  phases,  and  become  conscious  of 
each  separately,  he  can  even  count  them  without  difficulty.*  But  when  the  interval 
between  the  two  tones  increases  to  about  a  Semitone,  the  number  of  beats  becomes 
20  or  30  in  a  second,  and  the  ear  is  conse([uently  unable  to  follow  them  sufficiently 
well  for  counting.  If,  however,  we  begin  with  hearing  slow  beats,  and  then  increase 
their  rapidity  more  and  more,  we  cannot  fail  to  recognise  that  the  sensational  im- 
pression on  the  ear  preserves  precisely  the  same  character,  appearing  as  a  series 
of  separate  pulses  of  sound,  even  when  their  frequency  is  so  great  that  we  have 

H  no  longer  time  to  fix  each  beat,  as  it  passes,  distinctly  in  our  consciousness  and 
count  it.f 

But  while  the  hearer  in  this  case  is  quite  capable  of  distinguishing  that  his  ear 
now  hears  30  beats  of  the  same  kind  as  the  4  or  6  in  a  second  which  he  heard 
before,  the  effect  of  the  collective  impression  of  such  a  rapid  beat  is  qiute  different. 
In  the  first  place  the  mass  of  tone  becomes  confused,  which  1  principally  refer  to 
the  psychological  impressions.  We  actually  hear  a  series  of  pulses  of  tone,  and 
are  able  to  recognise  it  as  such,  although  no  longer  capable  of  following  each 
singly  or  separating  one  from  the  other.  But  besides  this  psychological  considera- 
tion, the  sensible  impression  is  also  unpleasant.  Such  rapidly  beating  tones  are 
jarring  and  rough.  The  distinctive  property  of  jarring,  is  the  intermittent  cha- 
racter of  the  sound.  We  think  of  the  letter  R  as  a  chai'acteristic  example  of 
a  jarring  tone.  It  is  well  known  to  be  produced  by  interposing  the  uvula,  or  else 
the  thin  tip  of  the  tongue,  in  the  way  of  the  stream  of  air  passing  out  of  the  mouth, 

H  in  such  a  manner  as  only  to  allow  the  air  to  force  its  way  through  in  sepai'ate  pulses, 
the  consequence  being  that  the  voice  at  one  time  soiuids  freely,  and  at  another  is 
cut  off.X 

Intermittent  tones  were  also  produced  on  the  double  siren  just  described  by 
using  a  little  reed  pipe  instead  of  the  wind-conduit  of  the  upper  box,  and  driving 
the  air  through  this  reed  pipe.  The  tone  of  this  pipe  can  be  heard  externally  only 
when  the  revohition  of  the  disc  brings  its  holes  before  the  holes  of  the  box  and 
opens  an  exit  for  the  air.  Hence,  if  we  let  the  disc  revolve  while  air  is  driven 
through  the  pipe,  we  obtain  an  intermittent  tone,  which  sounds  exactly  like  beats 
arising  from  two  tones  sounded  at  once,  although  the  intermittence  is  produced  by 
purely  mechanical  means.  Such  effects  may  also  be  produced  in  another  way  on 
the  same  siren.  Remove  the  loAver  windbox  and  retain  only  its  pierced  cover, 
over  which  the  disc  revolves.  At  the  under  part  apply  one  extremity  of  an  india- 
rubber  tube  against  one  of  the  holes  in  the  cover,  the  other  end  being  conducted 

U  by  a  proper  ear-piece  to  the  observer's  ear.  The  revolving  disc  alternately  opens 
and  closes  the  hole  to  which  the  india-rubber  tube  has  been  applied.  Hold  a 
tuning-fork  in  action  or  some  other  suitable  musical    instrument  above  and  near 

*  [See  App.  XX.  sect.  B.  No.  7,  for  direc-  Octave,    but   become   rapidly  too   fast   to  be 

tions  for  observing  beats. — Travslator.]  followed.     As,  however,  these  are  not  simple 

t  [The  Harmonical  is  very  convenient  for  tones,    the    beats   are   not    perfectly   clear. — 

this   purpose.      On   the   (l\y  key  is  a  f/j    one  Translator.'] 

comma  lower  than  d.  These  dd^  beat  about  \  [Phonautographic  figures  of  the  effect 
9,  18,  36,  73  times  in  10  seconds  in  the  of  r,  resemble  those  of  fig.  58,  p.  166rt.  Six 
different  Octaves,  the  last  barely  countable.  varieties  of  these  figures  are  given  on  p.  19  of 
Also  e^\)  and  Cj  beat  38,  66,  132,364  in  10  Donder'simportantlittle  tract  on 'The  Physio- 
seconds  in  the  different  Octaves.  The  two  first  logy  of  Speech  Sounds,  and  especially  of  those 
of  these  sets  of  beats  can  be  counted,  the  two  in  the  Dutch  Language  '  [Dc  Pliysiologie  der 
last  cannot  be  counted,  but  will  be  distinctly  Spraakkhnikci),  in  hct  hijzonder  ran  die  der 
perceived  as  separate  pulses.  Similarly  the  nederlandsche  taal.  Utrecht  1870,  pp.  24), — 
beats  between  all  consecutive  notes  (except  F  Translator.'] 
and  G,  B  and  C)  can  be  counted  in  the  lowest 


CHAP.  vrir.  LIMITS  OF  THE  FREQUENCY  OF  BEATS.  169 

the  rotating  disc.  Its  tone  will  be  heard  intermittently  and  the  number  of 
intermissions  can  be  regidated  by  altering  the  velocity  of  the  rotation  of  the 
disc. 

In  both  ways  then  we  obtain  intermittent  tones.  In  the  first  case  the  tone  of 
the  reed  pipe  as  heard  in  the  outer  air  is  interrupted,  because  it  can  only  escape 
from  time  to  time.  The  intermittent  tone  in  this  case  can  be  heard  by  any  number 
of  listeners  at  once.  In  the  second  case  the  tone  in  the  outer  air  is  continuous, 
but  reaches  the  ear  of  the  observer,  who  hears  it  through  the  disc  of  the  siren, 
intennittently.  It  can  certainly  be  heard  by  one  observer  only,  but  then  all  kinds 
of  musical  tones  of  the  most  diverse  pitch  and  quality  may  be  employed  for  the 
purpose.  The  intermission  of  their  tones  gives  them  all  exactly  the  same  kind  of 
roughness  which  is  produced  by  two  tones  which  beat  rapidly  together.  We  thus 
come  to  recognise  clearly  that  beats  and  intermissions  are  identical,  and  that  either 
when  fast  enough  produces  what  is  termed  a  jar  or  rattle.  ^ 

Beats  produce  intermittent  excitement  of  certain  auditory  nerve  fibres.  The 
reason  why  such  an  intermittent  excitement  acts  so  much  more  unpleasantly  than 
an  equally  strong  or  even  a  stronger  continuous  excitement,  may  be  gathered  from 
the  analogous  action  of  other  human  nerves.  Any  powerful  excitement  of  a  nerve 
deadens  its  excitability,  and  consequently  renders  it  less  sensitive  to  fresh  irritants. 
But  after  the  excitement  ceases,  and  the  nerve  is  left  to  itself,  irritability  is  speedily 
re-established  in  a  living  body  by  the  influence  of  arterial  blood.  Fatigue  and  re- 
freshment apparently  supervene  in  different  organs  of  the  body  •^ith  different 
velocities  ;  but  they  are  found  wherever  muscles  and  nerves  have  to  operate.  The 
eye,  which  has  in  many  respects  the  greatest  analogy  to  the  ear,  is  one  of  those 
organs  in  which  both  fatigue  and  refreshment  rapidly  ensue.  We  need  to  look  at 
the  sun  but  an  instant  to  find  that  the  portion  of  the  retina,  or  nervous  expansion 
of  the  eye,  which  was  affected  by  the  solar  light  has  become  less  sensitive  for  other 
light.  Immediately  afterwards  on  turning  our  eyes  to  a  uniformly  illuminated  ^ 
surface,  as  the  sky,  we  see  a  dark  spot  of  the  apparent  size  of  the  sun  ;  or  several 
such  spots  with  lines  between  them,  if  we  had  not  kept  our  eye  steady  when  look- 
ing at  the  sun  but  had  moved  it  right  and  left.  An  instant  suffices  to  produce  this 
effect ;  nay,  an  electric  spark,  that  lasts  an  immeasurably  short  time,  is  fully 
capable  of  causing  this  species  of  fatigue. 

When  we  continue  to  look  at  a  bright  surface,  the  impression  is  strongest  at 
first,  but  at  the  same  time  it  blunts  the  sensibility  of  the  eye,  and  consequently 
the  impression  becomes  weaker,  the  longer  we  allow  the  eye  to  act.  On  coming 
out  of  darkness  into  full  daylight  we  feel  blinded  ;  but  after  a  few  minutes,  when 
the  sensibility  of  the  eye  has  been  blunted  by  the  irritation  of  the  light, — or,  as  we 
say,  when  the  eye  has  grown  accustomed  to  the  glare, — this  degree  of  brightness  is 
found  very  pleasant.  Conversely,  in  coming  from  full  daylight  into  a  dark  vault, 
we  are  insensible  to  the  weak  light  aboiit  us,  and  can  scarcely  find  our  way  about, 
yet  after  a  few  minutes,  when  the  eye  has  rested  from  the  effect  of  the  strong  light,  H 
we  are  able  to  see  very  well  in  the  semi-dark  room. 

These  phenomena  and  the  like  can  be  conveniently  studied  in  the  eye,  because 
individual  spots  in  the  eye  may  be  excited  and  others  left  at  rest,  and  the  sensations 
of  each  may  be  afterwards  compared.  Put  a  piece  of  black  paper  on  a  tolerably 
well-lighted  white  surface,  look  steadily  at  a  point  on  or  near  the  black  paper,  and 
then  withdraw  the  paper  suddenly.  The  eye  sees  a  secondary  image  of  the  black 
paper  on  the  white  surface,  consisting  of  that  portion  of  the  white  surface  where 
the  black  paper  lay,  which  now  appears  brighter  than  the  rest.  The  place  in  the 
eye  where  the  image  of  the  black  paper  had  been  formed,  has  been  rested  in  com- 
parison with  all  those  places  which  had  been  afi'ected  by  the  white  svu-face,  and 
on  removing  the  black  paper  this  rested  part  of  the  eye  sees  the  white  surface  in 
its  first  fresh  brightness,  while  those  parts  of  the  retina  which  had  been  already 
fatigued  by  looking  at  it,  see  a  decidedly  greyer  tinge  on  the  whiter  surface. 

Hence  bv  the  continuous  uniform  action  of  the  irritation  of  light,  this  irritation 


170  LIMITS  OF  THE  FREQUENCY  OF  BEATS.  part  ii. 

itself  blunts  the  sensibility  of  the  nerve,  and  thus  eftectually  protects  this  organ 
against  too  long  and  too  violent  excitement. 

It  is  quite  different  when  we  allow  intermittent  light  to  act  on  the  eye,  such  as 
flashes  of  light  Avitli  intermediate  pauses.  During  these  pauses  the  sensibility  is 
again  somewhat  re-established,  and  the  new  irritation  consequently  acts  much 
more  intensely  than  if  it  had  lasted  with  the  same  uniform  strength.  Every  one 
knows  how  unpleasant  and  annoying  is  any  flickering  light,  even  if  it  is  relatively 
very  weak,  coming,  for  example,  from  a  little  flickering  taper  or  rushlight. 

The  same  thing  holds  for  the  nerves  of  touch.  Scraping  with  the  nail  is  far 
more  annojdng  to  the  skin  than  constant  pressure  on  the  same  place  with  the 
same  pressure  of  the  nail.  The  unpleasantness  of  scratching,  rubbing,  tickling, 
depends  upon  the  intermittent  excitement  which  they  produce  in  the  nerves  of 
touch. 
^  A  jarring  intermittent  tone  is  for  the  nerves  of  hearing  what  a  flickering  light 
is  to  the  nerves  of  sight,  and  scratching  to  the  nerves  of  touch.  A  much  more 
intense  and  unpleasant  excitement  of  the  organs  is  thus  prodiiced  than  would  be 
occasioned  by  a  continuous  uniform  tone.  This  is  even  shewn  when  we  hear  very 
weak  intermittent  tones.  If  a  tuning-fork  is  struck  and  held  at  such  a  distance 
from  the  ear  that  its  sound  cannot  be  heard,  it  becomes  immediately  audible  if  the 
handle  of  the  fork  be  revolved  by  the  fingers.  The  revolution  brings  it  alternately 
into  positions  where  it  can  and  cannot  transmit  sound  to  the  ear  [p.  161/y],  and 
this  alternation  of  strength  is  immediately  perceptible  by  the  ear.  For  the  same 
reason  one  of  the  most-delicate  means  of  hearing  a  very  weak,  simple  tone  consists 
in  sounding  another  of  nearly  the  same  strength,  which  makes  from  2  to  4  beats  in 
a  second  with  the  first.  In  this  case  the  strength  of  the  tone  varies  from  nothing 
to  4  times  the  sti'ength  of  the  single  simple  tone,  and  this  increase  of  strength 
combines  with  the  alternation  to  make  it  audible. 
51  Just  as  this  alternation  of  strength  will  serve  to  strengthen  the  impression  of 
the  very  weakest  musical  tones  upon  the  ear,  we  must  conclude  that  it  must  also 
serve  to  make  the  impression  of  stronger  tones  much  xuore  penetrating  and  violent, 
than  they  would  be  if  their  loudness  wei-e  continuous. 

We  have  hitherto  confined  our  attention  to  cases  where  the  number  of  beats 
did  not  exceed  20  or  30  in  a  second.  We  saw  that  the  beats  in  the  middle  pai't  of 
the  scale  are  still  quite  audible  and  form  a  series  of  separate  pulses  of  tone.  But 
this  does  not  furnish  a  limit  to  their  nximber  in  a  second. 

The  interval  V  c"  gave  us  33  beats  in  a  second,  and  the  eflfect  of  sounding  the  two 
notes  together  was  very  jarring.  The  interval  of  a  whole  tone  h'\)  c"  gives  nearly 
twice  as  many  beats,  but  these  are  no  longer  so  cutting  as  the  former.  The  rule 
assigns  88  beats  in  a  second  to  the  minor  Third  a  c",  but  in  reality  this  interval 
scarcely  shews  any  of  the  roughness  produced  by  beats  from  tones  at  closer  intervals. 
We  might  then  be  led  to  conjecture  that  the  increasing  number  of  beats  weakened 
H  their  impression  and  made  them  inaudible.  This  conjecture  would  find  an  analogy 
in  the  impossibility  of  separating  a  series  of  rapidly  succeeding  impressions  of 
light  on  the  eye,  when  their  number  in  a  second  is  too  lai-ge.  Think  of  a  glowing 
stick  swung  round  in  a  circle.  If  it  executes  10  or  15  revolutions  in  a  second,  the 
eye  believes  it  sees  a  continuous  circle  of  fire.  Similarly  for  colour-tops,  with 
which  most  of  my  readers  are  probably  familiar.  If  the  top  be  spun  at  the  rate 
of  more  than  10  revolutions  in  a  second,  the  colours  upon  it  mix  and  form  a  per- 
fectly unchanging  impression  of  a  mixed  colour.  It  is  only  for  very  intense  light 
that  the  alternations  of  the  various  fields  of  colour  have  to  take  place  more  quickly, 
20  to  30  times  in  a  second.  Hence  the  phenomena  are  ipiite  analogous  for  ear  and 
eye.  When  the  alternation  between  irritation  and  rest  is  too  fast,  the  alternation 
ceases  to  be  felt,  and  sensation  becomes  continuous  and  lasting. 

However,  we  may  convince  ourselves  that  in  the  case  of  the  ear,  an  increase  of 
the  number  of  beats  in  a  second  is  not  the  only  cause  of  the  disappearance  of  the 


CHAP.  viii.  LIMITS  OF  THE  FREQUENCY  OF  BEATS.  171 

corresponding  sensation.  As  we  passed  from  the  interval  of  a  Semitone  U  c"  to 
that  of  a  minor  Third  (i  c",  we  not  only  inci-eased  the  number  of  beats,  but  the 
width  of  the  interval.  Now  we  can  increase  the  number  of  beats  without  inci-easing 
the  interval  by  taking  it  in  a  higher  Octave.  Thus  taking  //  c"  an  Octave  higher 
we  have  b"  c"  with  66  beats,  and  another  Octave  would  give  us  //"  c""  with  as 
many  as  132  beats,  and  these  ai'e  really  audible  in  the  same  way  as  the  33  beats 
of  b'  c",  although  they  certainly  become  weaker  in  the  higher  positions.  Never- 
theless the  6Q  beats  of  the  interval  b"  c"  are  much  more  distinct  and  penetrating 
than  the  same  number  in  the  whole  Tone  b'\}  c",  and  the  88  of  the  interval  e" /'" 
are  still  quite  evident,  while  the  88  of  the  minor  Third  a  c"  are  practically  in- 
audible. My  assertion  that  as  many  as  132  beats  in  a  second  are  audible  will  per- 
haps appear  very  strange  and  incredible  to  acousticians.  But  the  ex})criment  is 
easy  to  repeat,  and  if  on  an  instrument  which  gives  sustained  tones,  as  an  organ 
or  harmonium,  we  strike  a  series  of  intervals  of  a  Semitone  each,  beginning  low  II 
•down,  and  proceeding  higher  and  higher,  we  shall  hear  in  the  lower  parts  very 
slow  beats  (B^  C  gives  4i,  B  c  gives  8i,  b  c  gives  16|  beats  in  a  second),  and  as  we 
ascend  the  rapidity  will  increase  but  the  character  of  the  sensation  remain  im- 
altered.  And  thus  we  can  pass  gradually  from  4  to  132  beats  in  a  second,  and 
-convince  ourselves  that  though  we  become  incapable  of  counting  them,  their  cha- 
racter as  a  series  of  pulses  of  tone,  producing  an  intermittent  sensation,  remains 
unaltered.  It  must  be  observed,  however,  that  the  beats,  even  in  the  higher  parts 
of  the  scale,  become  much  shriller  and  more  distinct,  when  their  iiumber  is 
diminished  by  taking  intervals  of  quarter  tones  or  less.  The  most  penetrating 
roughness  arises  even  in  the  upper  parts  of  the  scale  from  beats  of  30  to  40  in  a 
second.  Hence  high  tones  in  a  chord  are  much  more  sensitive  to  an  error  in 
tuning  amounting  to  the  fraction  of  a  Semitone,  than  deep  ones.  While  two  c 
notes  which  differ  from  one  another  by  the  tenth  part  of  a  Semitone,  produce  about 
3  beats  in  two  seconds,*  which  cannot  be  observed  without  considerable  attention,  ^ 
and,  at  least,  give  no  feeling  of  roughness,  two  c"  notes  with  the  same  error  give 
3  beats  in  one  second,  and  two  c"  notes  6  beats  in  one  second,  which  become  very 
disagreeable.  The  character  of  the  roughness  also  alters  with  the  number  of  beats. 
Slow  beats  give  a  coarse  kind  of  roughness,  which  may  be  considered  as  chattering 
or  jarring  ;  and  quicker  ones  have  a  finer  but  more  cutting  roughness. 

Hence  it  is  not,  or  at  least  not  solely,  the  large  nimiber  of  beats  which  renders 
them  inaiidible.  The  magnitude  of  the  interval  is  a  factor  in  the  result,  and  con- 
sequently we  are  able  with  high  tones  to  produce  more  rapid  audible  beats  than 
with  low  tones. 

Observation  shews  us,  then,  on  the  one  hand,  that  equally  large  intervals  by 
no  means  give  equally  distinct  beats  in  all  parts  of  the  scale.  The  increasing 
number  of  beats  in  a  second  renders  the  beats  in  the  upper  part  of  the  scale  less 
■distinct.  The  beats  of  a  Semitone  remain  distinct  to  the  upper  limits  of  the  four- 
times  accented  octave  [say  4000  vib.],  and  this  is  also  about  the  limit  for  musically 
tones  fit  for  the  combinations  of  harmony.  The  beats  of  a  whole  tone,  which  in 
■deep  positions  are  very  distinct  and  powerful,  are  scarcely  audible  at  the  upper 
limit  of  the  thi-ice-accented  octave  [say  at  2000  vib.].  The  major  and  minor 
Third,  on  the  other  hand,  which  in  the  middle  of  the  scale  [264  to  528  vib.]  may 
be  regarded  as  consonances,  and  when  justly  intoned  scarcely  shew  any  roughness, 
<are  decidedly  rough  in  the  lower  octaves  and  produce  distinct  beats. 

On  the  other  hand  we  have  seen  that  distinctness  of  beating  and  the  roughness 
of  the  combined  sounds  do  not  depend  solely  on  the  number  of  beats.  For  if  we 
■could  disregard  their  magnitudes  all  the  following  intervals,  which  by  calculation 
should  have  33  beats,  would  be  equally  rough  : 

*  [Taking  c'  =  264,  a  tone  one-tenth  of  a  second.  The  figures  in  the  text  have  been 
Semitone  or  10  cents  higher  make  265 '5  vibra-  altered  to  these  more  exact  numbers. —  I'rans- 
tions,  and   these   tones    beat    1^   times   in   a       lator.] 


172  LIMITS  OF  THP:  FREQUENCY  OF  BEATS.  taht  ii. 

the  Semitone  .  .     b'  c"  [528-495  =  33] 

the  whole  Tones      .  .     e  J    [major,  297-26-1]  and  d'  e  [minor  330-297] 

the  minor  Third      .  •     ^  i)     [198-165] 

the  major  Third      .  .     c  e      [165-132] 

the  Fourtli      .         .         .     G  c    [132-99] 
the  Fifth         .         .         .     C  G  [99-66] 
and  yet  we  find  that  these  intervals  are  more  and  more  free  from  rouglmess.* 

The  roughness  arising  from  sounding  two  tones  together  depends,  then,  in  a 
compound  manner  on  the  magnitude  of  the  interval  and  the  number  of  beats  pro- 
duced in  a  second.  On  seeking  for  the  reason  of  this  dependence,  we  observe  that, 
as  before  remarked,  beats  in  the  air  can  exist  only  when  two  tones  are  produced 
sufficiently  near  in  the  scale  to  set  the  same  elastic  appendages  of  the  auditory 
nerve  in  sympathetic  vibration  at  the  same  time.  When  the  two  tones  produced 
^  are  too  far  apart,  the  vibrations  excited  by  both  of  them  at  once  in  Corti's  organs 
are  too  weak  to  admit  of  their  beats  being  sensibly  felt,  supposing  of  course  that 
no  upper  partial  or  combinational  tones  intervene.  According  to  the  assumptions 
made  in  the  last  chapter  respecting  the  degree  of  damping  possessed  by  Corti's 
organs  (p.  144c),  it  would  result,  for  example,  that  for  the  interval  of  a  whole  Tone 
c  cZ,  such  of  Corti's  fibres  as  have  the  proper  tone  fTi,  would  be  excited  by  each  of 
the  tones  with  y\j  of  its  own  intensity  ;  and  these  fibres  will  therefore  fluctuate 
between  the  intensities  of  vibration  0  and  -~^.  But  if  we  strike  the  simple  tones  e 
and  cfi,  it  follows  from  the  table  there  given  that  Corti's  fibres  which  correspond 
to  the  middle  between  c  and  (A  will  alternate  between  the  intensities  0  and  \^. 
Conversely  the  same  intensity  of  beats  would  for  a  minor  Third  amount  to  only 
0*194,  and  for  a  major  Third  to  only  0*108,  and  hence  would  be  scarcely  perceptible 
beside  the  two  pi-imary  tones  of  the  intensity  1. 

Fig.  59,  which  we  used  on  p.  144f/  to  express  the  ^"^-  ■''^• 

H  intensity  of  the  sympathetic  vibration  of  Corti's  ti 

fibres  for  an  increasing  interval  of  tone,  may 
here  serve  to  shew  the  intensity  of  the  beats 
which  two  tones  excite  in  the  ear  when  forming 
different  intervals  in  the  scale.  But  the  parts  on 
the  base  line  must  now  be  considered  to  repre-       ^  ,_.^-^'\\ 

sent  fifths  of  a  whole  Tone,  and  not  as  before  of        lo  so  5  lo 

a  Semitone.     In  the  present  case  the  distance  of 

the  two  tones  from  each  other  is  doubly  as  great  as  that  between  either  of  them 
and  the  intermediate  Corti's  fibres. 

Had  the  damping  of  Corti's  organs  been  equally  great  at  all  parts  of  the  scale, 
and  had  the  number  of  beats  no  influence  on  the  roughness  of  the  sensation,  equal 
intervals  in  all  parts  of  the  scale  woidd  have  given  equal  roughness  to  the  combined 
tones.  But  as  this  is  not  the  case,  as  the  same  intervals  diminish  in  roughness. 
%  as  we  ascend  in  the  scale,  and  increase  in  roughness  as  we  descend,  we  must  either 
assume  that  the  damping  power  of  Corti's  organs  of  higher  pitch  is  less  than  that 
of  those  of  lower  pitch,  or  else  that  the  discrimination  of  the  more  rapid  beats 
meets  with  certain  hindrances  in  the  nature  of  the  sensation  itself. 

At  present  I  see  no  way  of  deciding  between  these  two  suppositions ;  but  the 
former  is  possibh-  the  more  improbable,  because,  at  least  with  our  artificial  musical 
instruments,  the  higher  the  pitch  of  a  vibrating  body,  the  more  difficulty  is  ex- 
perienced in  isolating  it  sufficiently  to  prevent  it  from  commiuiicating  its  vibrations 
to  its  environment.  Very  short,  high-pitched  strings,  little  metal  tongues  or  plates, 
&c.,  yield  high  tones  which  die  off  with  great  rapidity,  whereas  it  is  easy  to 
generate  deep  tones  with  correspondingly  greater  bodies  which  shall  retain  their 
tone  for  a  considerable  time.  On  the  other  hand  the  second  supposition  is  favoured 
by  the  analogy  of  another  nervous  apparatus,  the  eye.     As  has  been  already  re- 

*  [All  these  intervals  can  be  tried  on  the  the  student  should  listen  to  the  beats  of  the 
Harmonical,  but  as  the  tones  are  compoimd,       primes  only. — Translator.'] 


CHAP.  vm.  LIMITS  OF  THE  FREQUENCY  OF  BEATS.  173 

marked,  a  series  of  impressions  of  light,  following  each  other  rapidly  and  re'^ularlv 
excite  a  uniform  and  continuous  sensation  of  light  in  the  eve.  ^V^hen  the  separate 
luminous  in-itations  follow  one  another  very  quickly,  the  impression  produced  by 
each  one  lasts  unweakened  in  the  nerves  till  the  next  supervenes,  and  thus  the 
pauses  can  no  longer  be  distinguished  in  sensation.  In  the  eye,  the  number  of 
.separate  irritations  cannot  exceed  24:  in  a  second  without  being  completely  fused 
into  a  single  sensation.  In  this  respect  the  eye  is  far  surpassed  by  the  ear,  which 
can  distinguish  as  many  as  132  intermissions  in  a  second  and  probably  even  that 
is  not  the  extreme  limit.  Much  higher  tones  of  sutiicient  strength  would  probably 
allow  us  to  hear  still  more.*  It  lies  in  the  nature  of  the  thing,  that  different  kind's 
of  apparatus  of  sensation  should  shew  different  degrees  of  mobility  in  this  respect, 
since  the  result  does  not  depend  simply  on  the  mobility  of  the  molecules  of  the 
nerves,  but  also  depends  upon  the  mobility  of  the  auxiliary  apparatus  through 
which  the  excitement  is  induced  or  expressed.  Muscles  are  much  less  mobile  thanH 
the  eye  ;  ten  electrical  discharges  in  a  second  directed  through  them  generally 
suffice  to  bring  the  voluntary  muscles  into  a  permanent  state  of  contraction.  For 
the  muscles  of  the  involuntary  system,  of  the  bowels,  the  vessels,  &c.,  the  pauses 
between  the  irritations  may  be  as  much  as  one,  or  even  several  seconds  long,  with- 
out any  intermission  in  the  continuity  of  contraction. 

The  ear  is  greatly  superior  in  this  respect  to  any  other  nervous  apparatus.  It 
is  eminently  the  organ  for  small  intervals  of  time,  and  has  been  long  used  as  such 
by  astronomers.  It  is  well  known  that  when  two  pendulums  are  ticking  near  one 
another,  the  ear  can  distinguish  whether  the  ticks  are  or  are  not  coincident,  within 
one  hundredth  of  a  second.  The  eye  would  certainly  fail  to  determine  whether 
two  flashes  of  light  coincided  within  ^/^  second  ;  and  probably  within  a  much  larger 
fraction  of  a  second.! 

But  although   the   ear  shews  its  superiority  over  other  organs  of  the  body  in 
this  respect,  we  cannot  hesitate  to  assume  that,  like  every  other  nervous  apparatus,  ^ 
the  rapidity  of  its  power  of  apprehension  is  limited,  and  we  may  even  assume  that 
we  have  approached  very  near  the  limit  when  we   can  but  faintly  distinguish  132 
Iteats  in  a  second. 

*  [In  tlie  two  high  notes  g""  f'jjf  of  the  appear  beliind  a  bar,  and  an  electrical  current 

flageolet  fifes  (p.   153(Z,  note),  which  if  justly  causes  another  point  to  make  a  hole  between 

intoned  should  give  198  beats  in  a  second,  I  the  seconds  holes  on    the  chronograph.      By 

could  hear  none,  though  the  tones  were  very  applying  a  scale,  the  time  of  transit  is  thus 

powerful,    and   the   scream   was  very   cutting  measured  off.     A  mean,  of  course,  is  taken  as 

indeed.— In  the  case  of  b"  c'",  which  on  the  before.     On  my  asking  Mr.  Stone  (now  Astrono- 

Harmonical  are  tuned  to  make  1056  and  990,  mer  at  Oxford,  then  chief  assistant  at  Green- 

tlie  rattle  of  the  66  beats,  or  thereabouts,  is  wich  Observatory)  as  to  the  relative  degree  of 

quite  distinct,  and  the  differential  tone  is  very  accuracy  of  the  two  methods,  he  told  me  that 

powerful  at  the  same  time.— 2Va?i.s/rtfi!or.]  he  considered  the   first  gave  results   to  one- 

t  [The  following  is  an  interesting  compari-  tenth,  and  the  second  to  one-twentieth  of  a 

son  between  eye  and   ear,  and  eye  and  hand.  second.     It  must  be  remembered  that  the  first 

The  usual  method  of  observing  transits  is  by  method   also  required    a   mental    estimation 

counting  the  pendulum  ticks  of  an  astronomi-  which   had  to    be   performed   in   less  than  a 

cal  clock,  and  by  observing  the  distances  of  second,  and  the  result  borne  in  mind,  and  that  % 

the  apparent  positions  of  a  star  before  and  after  this  was  avoided  by  the  second  plan.     On  the 

passing  each  bar  of  the  transit  instrument  at  other  hand  in  the  latter  the  sensation  had  to 

the  moments  of  ticking,  to  estimate  the  moment  be  conveyed  from  the  eye  to  the  brain,  which 

at  which  it  had  passed  each  bar.     This  is  done  issued  its  orders  to  the  hand,  and  the  hand 

for  five  bars  and  a  mean  is  taken.     But  a  few  had  to  obey  them.  Hence  there  was  an  endea-    ■ 

years  ago  a  chronograph  was   introduced  at  vour   at   performing    simultaneously,    several 

Greenwich   Observatory,  consisting  of  a  uni-  acts  which  could  only  be  successive.     Anyone 

formly   revolving  cylinder   in  which  a  point  will  find  upon  trial  that  an  attempt  to  merely 

pricks  a  hole  every  second.     Electrical  com-  make  a  mark  at  the  moment   of  hearing  an 

munication  being  established  with  a  knob  on  expected  sound,  as,  for  example,  the  repeated 

the  transit  instrument,   the  observer  presses  tick  of  a  common  half  seconds  clock,  is  liable 

the  knob  at  the  moment  he  sees  a  star  dis-  to  great  error. — Translator.] 


174  DEEP  AND  DEEPEST  TONES.  part  ik 

CHAPTER  IX. 

DEEP    AND    DEEPEST    TONES. 

Beats  give  us  an  important  means  of  determining  the  limit  of  the  deepest  tones, 
and  of  accounting  for  certain  peculiarities  of  the  transition  from  the  sensation  of 
separate  pulses  of  air  to  a  perfectly  continuous  musical  tone,  and  to  this  inquiry 
\vc  now  proceed. 

The  question  :  what  is  the  smallest  number  of  vibrations  in  a  second  which 
can  produce  the  sensation  of  a  musical  tone  1  has  hitherto  received  very  contra- 
dictory replies.     The  estimates  of  different  observers  fluctuate  between  8  (Savart) 

<^  and  about  30.  The  contradiction  is  explained  by  the  existence  of  certain  difficul- 
ties in  the  experiments. 

In  the  first  place  it  is  necessary  that  the  strength  of  the  vibrations  of  the  air 
for  very  deep  tones  should  be  extremely  greater  than  for  high  tones,  if  they  are  to 
make  as  strong  an  impression  on  the  ear.  Several  acousticians  have  occasionally 
started  the  hypothesis  that,  caeteris  paribtLs,  the  strength  of  tones  of  diflferent 
heights  is  directly  proportional  to  the  vis  viva  of  the  motion  of  the  air,  or,  which 
comes  to  the  same  thing,  to  the  amount  of  the  mechanical  work  applied  for  pro- 
ducing it.  But  a  simple  experiment  with  the  siren  shews  that  when  equal  amounts 
of  mechanical  work  are  applied  to  produce  deep  and  high  tones  under  conditions 
otherwise  alike,  the  high  tones  excite  a  very  much  more  powerful  sensation  than 
the  deep  ones.  Thus,  if  the  siren  is  blown  by  a  bellows,  which  makes  its  disc 
revolve  with  increasing  rapidity,  and  if  we  take  care  to  keep  up  a  perfectly' 
uniform  motion  of  the  bellows  by  raising  its  handle  by  the  same  amount  the  same 

^  number  of  times  in  a  minute,  so  as  to  keep  its  bag  equally  filled,  and  drive  the 
same  amount  of  air  under  the  same  pressure  through  the  siren  in  the  same  time, 
we  hear  at  first,  while  the  revolution  is  slow,  a  weati  deep  tone,  which  continually 
ascends,  but  at  the  same  time  gains  strength  at  an  extraordinary  rate,  till  when  the 
highest  tones  producible  on  my  double  siren  (near  to  a",  with  880  vibrations  in  a 
second)  are  reached,  their  strength  is  almost  insupportable.  In  this  case  by  far 
the  greatest  part  of  the  uniform  mechanical  work  is  applied  to  the  generation  of 
sonorous  motion,  and  only  a  small  part  can  be  lost  by  the  friction  of  the  revolving 
disc  on  its  axial  supports,  and  the  air  which  it  sets  into  a  vortical  motion  at  the 
same  time  ;  and  these  losses  must  even  be  greater  for  the  more  rapid  rotation  than 
for  the  slower,  so  that  for  the  production  of  the  high  tones  less  mechanical  work 
remains  applicable  than  for  the  deep  ones,  and  yet  the  higher  tones  appear  to  our 
sensation  extraordinarily  more  jjowerful  than  the  deep  ones.  How  far  upwards 
this  increase  may  extend,  I  have  as  yet  been  imable  to  determine,  for  the  rapidity 
of  my  siren  cannot  be  further  increased  with  the  same  pressure  of  air. 

^  The  increase  of  strength  with  height  of  tone  is  of  especial  consequence  in  the 
deepest  part  of  the  scale.  It  follows  that  in  compound  tones  of  great  depth,  the 
upper  partial  tones  may  be  superior  to  the  prime  in  strength,  even  though  in 
musical  tones  of  the  same  description,  but  of  greater  height,  the  strength  of  the 
prime  greatly  predominates.  This  is  readily  proved  on  my  double  siren,  because 
by  means  of  the  beats  it  is  easy  to  determine  whether  any  partial  tone  which  we 
hear  is  the  prime,  or  the  second  or  third  partial  tone  of  the  compound  under 
examination.  For  when  the  series  of  12  holes  are  open  in  both  windboxes,  and 
the  handle,  which  moves  the  upper  windbox,  is  rotated  once,  we  shall  have,  as 
already  shewn,  4  beats  for  the  primes,  8  for  the  second  partials,  and  12  for  the 
third  partials.  Now  we  can  make  the  disc  revolve  more  slowly  than  usual,  by 
allowing  a  well-oiled  steel  spring  to  rub  against  the  edge  of  one  disc  with  difterent 
degrees  of  pressure,  and  thus  we  can  easily  produce  series  of  puffs  which  corre- 
spond to  very  deep  tones,  and  then,  tiirning  the  handle,  we  can  count  the  beats. 


CHAP.  IX.  DEEP  AND  DEEPEST  TONES.  175 

By  allowing  the  rai)itlity  of  the  revolution  of  the  discs  to  increase  gradually,  wo 
find  that  the  first  audible  tones  produced  make  12  beats  for  each  revolution  of  the 
handle,  the  number  of  puffs  being  from  36  to  40  in  the  second.  Eor  tones  with 
from  40  to  80  puffs,  each  revolution  of  the  handle  gives  8  beats.  In  this  case, 
then,  the  upper  Octave  of  the  prime  is  the  strongest  tone.  It  is  not  till  we  have 
80  puff's  in  a  second  that  we  hear  the  four  beats  of  the  primes. 

It  is  proved  by  these  experiments  that  motions  of  the  air,  which  do  not  take 
the  form  of  pendular  vibrations,  can  excite  distinct  and  powerful  sensations  of  tone, 
of  which  the  pitch  number  is  2  or  3  times  the  number  of  the  pulses  of  the  air, 
and  yet  that  the  prime  tone  is  not  heard  through  them.  Hence,  when  we  continu- 
ally descend  in  the  scale,  the  strength  of  our  sensation  decreases  so  rapidly  that 
the  sound  of  the  prime  tone,  although  its  vis  viva  is  independently  greater  than  that 
of  the  upper  partials,  as  is  shewn  in  higher  positions  of  a  musical  tone  of  the 
same  composition,  is  overcome  and  concealed  by  its  own  upper  partials.  Even  H 
when  the  action  of  the  compound  tone  on  the  ear  is  much  reinforced,  the  effect 
remains  the  same.  In  the  experiments  with  the  siren  the  uppermost  plate  of  the 
bellows  is  violently  agitated  for  the  deep  tones,  and  when  I  laid  my  head  on  it,  my 
whole  head  was  set  into  such  powerful  sympathetic  vibration  that  the  holes  of  the 
rotating  disc,  which  vanish  to  an  eye  at  rest,  became  again  separately  visible, 
through  an  optical  action  similar  to  that  which  takes  place  in  stroboscopic  discs. 
The  row  of  holes  in  action  appeared  to  stand  still,  the  other  rows  seemed  to  move 
partly  backwards  and  partly  forwards,  and  yet  the  deepest  tones  were  no  more 
distinct  than  before.  Ac  another  time  I  connected  my  ear  by  means  of  a  properly 
introduced  tiibe  with  an  opening  leading  to  the  interior  of  the  bellows.  The 
agitation  of  the  drumskin  of  the  ear  was  so  great  that  it  produced  an  intolerable 
itching,  and  yet  the  deepest  tones  remained  as  indistinct  as  ever. 

In  order,  then,  to  discover  the  limit  of  deepest  tones,  it  is  necessary  not  only  to 
produce  very  violent  agitations  in  the  air  but  to  give  these  the  form  of  simple^ 
pendular  vibrations.  Until  this  last  condition  is  fulfilled  we  cannot  possibly  say 
whether  the  deep  tones  we  hear  belong  to  the  prime  tone  or  to  an  upper  partial  tone 
of  the  motion  of  the  air.*  Among  the  instruments  hitherto  employed  the  wide- 
stopped  organ  pipes  are  the  most  suitable  for  this  purpose.  Their  upper  partial 
tone^  are  at  least  extremely  weak,  if  not  quite  absent.  Here  we  find  that  even  the 
lower  tones  of  the  16-foot  octave,  C^  to  U^,  begin  to  pass  over  into  a  droning  noise, 
so  that  it  becomes  difficult  for  even  a  practised  musical  ear  to  assign  their  pitch  with 
certainty  ;  and,  indeed,  they  cannot  be  tuned  by  the  ear  alone,  but  only  indirectly 
by  means  of  the  beats  which  they  make  with  the  tones  of  the  upper  octaves.  We 
observe  a  similar  effect  on  the  same  deep  tones  of  a  piano  or  harmonium ;  they 
form  drones  and  seem  out  of  tune,  although  their  musical  character  is  on  the 
whole  better  established  than  in  the  pipes,  because  of  their  accompanying  upper 
partial  tones.  In  music,  as  artistically  applied  in  an  orchestra,  the  deepest  tone 
used  is,  therefore,  the  F^,  of  the  [4-stringed  German]  double  bass,  with  41}  vibra-H 
tions  in  a  second  [see  p.  18c,  note],  and  I  think  I  may  predict  with  certainty  that  all 
efforts  of  modern  art  applied  to  produce  good  musical  tones  of  a  lower  pitch  must 
fail,  not  because  proper  means  of  agitating  the  air  cannot  be  discovered,  but 
because  the  human  ear  cannot  hear  them.  The  16-foot  C^  of  the  organ,  with 
33  vibrations  in  a  second,  certainly  gives  a  tolerably  continuous  sensation  of 
drone,  but  does  not  allow  us  to  give  it  a  definite  position  in  the  musical  scale. 
We  almost  begin  to  observe  the  separate  pulses  of  air,  notwithstanding  the  regular 
form  of  the  motion.  In  the  upper  half  of  the  32-foot  octave,  the  perception  of  the 
separate  pulses  becomes   still   clearer,  and  the   continuous  part  of  the   sensation, 

*Thus  Savart's  instrument,  where  a  rota-  tion,  and  consequently  the  upper  partial  tones 

ting  rod  strikes  through  a  narrow  slit,  is  totally  must    be   very   strongly   developed,    and   the 

unsuitable  for  making  the  lowest  tone  audible.  deepest  tones,  which  are  heard  for  8  to  16 

The  separate  puffs  of  air  are  here  very  short  in  passages  of  the  rod  through  the  bole  in  a  second, 

relation  to  the  whole  periodic  tinre  of  the  vibra-  can  be  notliing  but  upper  partials. 


176 


DEEP  AND  DEEPEST  TONES. 


which  may  be  compared  with  a  sensation  of  tone,  continual!}'  weaker,  and  in  the 
lower  half  of  the  32-foot  octave  we  can  scarcely  be  said  to  hear  anything  but  the 
individual  pulses,  or  if  anything  else  is  really  heard,  it  can  only  be  weak  upper 
partial  tones,  from  which  the  musical  tones  of  stopped  pipes  are  not  quite  free. 

I  have  tried  to  produce  deep  simple  tones  in  another  way.  Strings  which  are 
weighted  in  their  middle  with  a  heavy  piece  of  metal,  on  being  struck  give  a  com- 
pound tone  consisting  of  many  simple  tones  which  are  mutually  inharmonic.  The 
prime  tone  is  separated  from  the  nearest  upper  pai'tials  by  an  interval  of  several 
Octaves,  and  hence  there  is  no  danger  of  confusing  it  with  any  of  them ;  besides, 
the  vipper  tones  die  away  rapidly,  but  the  deeper  ones  continue  for  a  very  long  time. 
A  string  of  this  kind  *  was  stretched  on  a  sounding-box  having  a  single  opening 
which  covild  be  connected  with  the  auditory  passage,  so  that  the  air  of  the  sounding- 
box  could  escape  nowhere  else  but  into  the  ear.  The  tones  of  a  string  of  customary 
51  pitch  are  under  these  circumstances  insupportably  loud.  But  for  D^,  with  37|- 
vibrations  in  a  second,  there  was  only  a  very  weak  sensation  of  tone,  and  even  this 
was  rather  jarring,  leading  to  the  conclusion  that  the  ear  began  even  here  to  feel 
the  separate  pulses  separately,  notwithstanding  their  regularity.  At  B^)y,  with 
29 J  vibrations  in  a  second,  there  was  scarcely  anj^thing  audible  left.  It  appears, 
then,  that  those  nerve  fibres  which  perceive  such  tones  begin  as  early  as  at  this 
note  to  be  no  longer  excited  with  a  uniform  degree  of  strength  during  the  whole 
time  of  a  vibration,  whether  it  be  the  phases  of  greatest  velocity  or  the  phases  of 
greatest  deviation  from  their  mean  position  in  the  vibrating  formations  in  the  ear 
which  eflfect  the  excitement. t 


*  It  was  a  thin  brass  pianoforte  string.  The 
weight  was  a  copper  kreutzer  piece  [pronounce 
kroitser :  three  kreutzers  make  a  penny  at 
Heidelberg,  where  the  expei-iment  was  pro- 
bably tried] ,  pierced  in  the  middle  by  a  hole 
■^  through  which  the  wire  passed,  and  then  made 
to  grip  the  wire  immovably  by  driving  a  steel 
point  between  the  hole  in  the  kreutzer  and  the 
string. 

t  Subsequently  I  obtained  two  large  tuning- 
forks  from  Herr  Koenig  in  Paris,  with  sliding 
weights  on  their  prongs.  By  altering  the  posi- 
tion of  the  weights,  the  jiitch  was  changed, 
and  the  corresponding  number  of  vibrations 
was  given  on  a  scale  which  runs  along  the 
prongs.  One  fork  gave  24  to  35,  the  other  35 
to  61  vibrations.  The  sliding  weight  is  a  plate, 
5  centimetres  [nearly  2  inches]  in  diameter, 
and  forms  a  mirror.  On  bringing  the  ear  close 
to  these  plates  the  deep  tones  are  well  heard. 
For  .30  vibrations  I  could  still  hear  a  weak 
drone,  for  28  scarcely  a  trace,  although  this 
arrangement  made  it  easily  possible  to  form 
^  oscillations  of  9  millimetres  [about  I  inch]  in 
amplitude,  quite  close  to  the  ear.  Prof.  W. 
Preyer  has  been  thus  able  to  hear  down  to  24 
vib.  He  has  also  applied  another  method 
( Physiologische  A  hhandhingcn,  Physiological 
Treatises,  Series  1,  part  1,  '  On  the  limits  of 
the  perception  of  tone,'  pp.  1-17)  by  using  very 
deep,  loaded  tongues,  in  reed  pipes,  which  were 
constructed  for  this  purpose  by  Herr  Appunn 
of  Hanau,  and  gave  from  8  to  40  vib.  These 
were  set  into  strong  vibration  by  blowing,  and 
then  on  interrupting  the  wind,  the  dying  off 
of  the  vibrations  was  listened  to  by  laying  the 
ear  against  the  box.  He  states  that  tones  were 
heard  downwards  as  low  as  15  vib.  But  the 
proof  that  the  tones  heard  corresponded  with 
the  primes  of  the  pipes  depends  only  on  the 
fact,  that  the  pitch  gradually  ascended  as  they 
passed  over  into  the  tones  of  from  25  to  32 
vib.,  which  were  more  audible,  but  died  o2  more 


rapidly.  With  extensive  vibrations,  however, 
the  tongues  may  have  very  easily  given  their 
point  of  attachment  longitudinal  impulses  of 
double  the  frequency,  because  when  they 
reached  each  extremity  of  their  amplitude  they 
might  drive  back  the  point  of  attaclunent 
through  their  flexion,  whereas  in  the  middle 
of  the  vibration  they  would  draw  it  forward  by 
the  centrifugal  force  of  their  weight.  Since 
the  power  of  distinguishing  pitch  for  these 
deepest  tones  is  extremely  imperfect,  I  do  not 
feel  my  doubts  removed  by  the  judgment  of 
the  ear  when  the  estimates  are  not  checked  by 
tlie  counting  of  beats. 

[This  check  I  am  fortunately  able  to  supply. 
A  copy  of  the  instrument  used  by  Prof.  Preyer 
is  in  the  South  Kensington  ]\Iuseum.  It  con- 
sists of  an  oblong  box,  in  the  lower  jmrt  of 
which  are  the  loaded  hai'monium  reeds,  not 
attached  to  pipes,  but  vibrating  within  the  box, 
and  governed  by  valves  which  can  be  opened 
at  pleasure.  On  account  of  the  beats  between 
tongue  and  tongue  taking  place  in  strongly 
condensed  air,  they  are  accelerated,  and  the 
nominal  pitch,  obtained  by  counting  the  beats 
from  reed  to  reed,  is  not  quite  the  same 
as  the  actual  pitch  (see  App.  XX.  sect.  B. 
No.  6).  The  series  of  tones  is  supposed  to 
proceed  from  8  to  32  \\h.  by  differences  of  1 
vib.,  from  32  to  64  by  differences  of  2  vib.,  and 
from  64  to  128  by  differences  of  4  vibs.  In 
November  1879,  for  another  purpose,  I  deter- 
mined the  pitch  of  every  reed  by  Scheibler's 
forks  (see  App.  XX.  sect.  B.  No.  7),  by  means 
of  the  upper  partials  of  the  reeds.  For  Eeeds 
8,  9,  10,  11,  I  used  from  the  20th  to  the  30th 
partial,  but  I  consider  only  Reed  11  as  quite 
certain.  I  found  it  made  10'97  vib.  by  the  20th, 
and  10-95  by  both  the  21st  and  24th  partials. 
From  Reed  11  upwards  I  determined  every 
pitch,  in  many  cases  by  several  partials,  the 
result  only  differing  in  the  second  place  of 
decimals.     I  give  the  two  lowest  Octaves,  the 


DEEP  AND  DEEPEST  TONES. 


177 


Hence  ill  though  tones  of  24  to  28  vib.  have  been  heard,  notes  do  not  begin  to 
have  a  definite  pitch  till  about  40  vibi-ations  are  performed  in  a  second.  These 
facts  will  agree  with  the  hypothesis  concerning  the  elastic  appendages  to  the  audi- 
tory nerves,  on  remembering  that  the  dee])ly  intoned  fibres  of  Corti  may  be  set  in 
sympathetic  vibration  by  still  deeper  tones,  although  with  rapidly  decreasing 
strength,  so  that  sensation  of  tone,  but  no  discrimination  of  pitch,  is  possible.  If 
the  most  deeply  intoned  of  Corti's  fibres  lie  at  greater  intervals  from  each  other  in 
the  scale,  but  at  the  same  time  their  damping  power  is  so  great  that  every  tone 
which  corresponds  to  the  pitch  of  a  fibre,  also  pretty  strongly  affects  the  neighbour- 
ing fibres,  there  will  be  no  safe  distinction  of  pitch  in  their  vicinity,  but  it  will 
proceed  continuously  without  jumps,  while  the  intensity  of  the  sensation  must  at 
the  same  time  become  small. 

Whilst  simple  tones  in  the  upper  half  of  the   16-foot  octave  are  perfectly  con- 


only  pitches  of  interest  for  the  present  pur- 
pose, premising  that  I  consider  the  three  lowest 
pitches  (for  which  the  upper  partials  lay  too 


close  together)  and  the  highest  (which  had  a  ^ 
bad  reed)  to  be  very  uncertain. 


Nominal 
Actual  - 

8 
7-91 

9 

8-89 

10 

9-81 

11 
10-95 

12 
11-90 

13 
12-90 

14 
13-93 

15 

14-91 

16 
15-91 

Nominal 
Actual    - 

-  17 

-  16-90 

18 
17-91 

19 
18-89 

20 
19-91 

21 
20-91 

22 
21-91 

23 

22-88 

24 
23-97 

25 
24-92 

Nominal 
Actual   - 

-  26 

-  25-92 

27 
26-86 

28 
27-85  ' 

29 

-   28-84 

30 
29-77 

31 
30-63 

32 
31-47 

There  can  therefore  be  no  question  as  to  the 
real  pitch.  At  Prof.  Preyer's  request  I  ex- 
amined this  instrument  in  Oct.  1877,  and  he 
has  printed  my  notes  in  his  Akustische  Unter- 
si(chungf7i,'PY>.  6-8.  From  these  I  extract  the 
following  : — 

R  means  Reed,  and  R  21  ■•25  means  that  the 
two  reeds  21  and  25  were  sounded  together  and 
gave  beats. 

R  21-25,  beat  4  in  1  sec,  counted  for  20  sec. 
Hence  both  of  their  lowest  partials  must  have 
been  effective. 

R  20-'24,  beat  4  in  1  sec,  counted  for  10  sec. 

R  19- -23,  beat  4  in  1  sec,  counted  for  20  sec. 

R  17' -21,  same  beats. 

R  16--20,  same  beats  quite  distinctly. 

R  15  "19,  at  first  I  lost  the  beats, but  afterwards 
by  getting  R  15  well  into  action  before  R  19  was 
set  on,  and  keeping  on  pumping,  I  got  out  the  4 
in  a  second  quite  distinctly.  Hence  the  lowest 
partial  of  R  15  was  effective. 

R  15-17,  here  also  I  once  heard  4  in  a  sec, 
but  this  must  have  been  from  the  Octaves. 

R  14'  16,  I  was  quite  unable  to  distinguish 
anything  in  the  way  of  beats,  but  volleys  like  a 
fe.i  dejoieahout  a  second  in  length, butimj)Ossible 
to  count  accurately;  they  may  have  been  2  in  a 
se?.  and  I  counted  double.  At  the  same  time  I 
seemed  occasionally  to  hear  a  low  beat,  so  low  and 
gentle  that  I  could  not  count  it,  and  the  great 
exertion  of  pumping  the  bellows  full  enough  to 
keep  these  two  low  reeds  iu  action,  prevented 
accurate  observation. 

R  15  decidedly  seemed  flatter  than  R  13,  so 
that  I  could  have  onlv  heard  the  lowest  pai'tial 
of  R  15  and  the  Octave  of  R  13. 

On  sounding  R  14  and  R  15  separately,  I 
seemed  to  hear  from  each  a  very  low  tone,  in 
quality  more  like  a  differential  tone  than  any- 
thing else.  This  could  also  be  heard  even  with 
R  13  and  R  12,  below  the  thumps,  and  even  in 
R  11. 

At  R  8  I  heard  only  the  sishing  of  the  escape 
of  wind  from  the  reed,  8  times  in  a  second,  as 
well  as  I  co;ild  count,  and  I  also  heard  beats 
evidently  arising  from  the  higher  partials,  and 
also  8  in  a  second. 

At  R  9  there  was  the  same  kind  of  sishing  and 
equally  rapid  beats.  But  iu  addition  I  seemed 
to  hear  a  faint  low  tone. 


At  R  10  there  was  no  mistake  as  to  the  ex- 
istence of  such  a  musical  tone. 

At  R  11  and  R  12  it  was  still  more  distinct. 

At  R  13  the  tone  was  very  distinct  and  was 
quite  a  good  musical  tone  at  R  14,  but  the  sish 
was  still  audible.  Was  this  the  lowest  partial 
or  its  Octave  ? 

R  10  gave  quite  an  organ  tone,  nothing  like 
a  hum  or  a  differential,  but  tlje  sish  and  beats 
remain.     I  must  have  heard  the  lowest  partial,  ^ 
and  b}'  continual  pumping  I  was  able  to  keep  it 
in  my  ear. 

R  18--20  gave  beats  of  2  in  a  sec.  very  distinctly. 

Up  to  R  25  the  sish  could  be  heard  at  the 
commencement,  but  it  rapidly  disappeared.  It 
feels  as  if  the  tone  were  getting  gradually  into 
practice.  This  effect  continued  up  to  R  22,  after 
which  the  sish  was  scarcely  brought  out  at  all. 
In  fact  long  before  this  the  sish  was  made  only 
at  the  first  moment,  and  was  rather  a  bubble 
than  a  sish. 

In  listening  to  the  very  low  beats,  the  beats  of 
the  lowest  partials  as  such  could  not  be  separated 
from  the  general  mass  of  beats,  but  the  4  in  a  sec. 
were  quite  clear  from  R  15 '■19.  The  lowest  pair 
in  which  I  was  distinctly  able  to  hear  the  bell-like 
beat  of  the  lowest  partials  distinct  from  the 
general  crasli  was  R  SO' 34.  But  I  fancied  I 
heard  it  at  R  28 ••32. 

Prof.  Preyer  also,  in  the  same  place,  details  II 
his  experiments  with  two  enormous  tuning- 
forks  giving  13-7  and  18-6  vib.  The  former 
gave  no  musical  tone  at  all,  though  the  vibra- 
tions wei-e  visible  for  3  min.  and  were  dis- 
tinctly separable  by  touch.  The  latter  had 
'an  unmistakable  dull  tone,  without  droning 
or  jarring  '.  He  concludes  :  '  Less  than  15 
vib.  in  a  sec.  give  no  musical  tone.  At  from 
16  to  24,  say